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| Boilerpipe Text | A visualisation of a solution to the two-dimensional
heat equation
with temperature represented by the vertical direction and color
In
mathematics
, a
partial differential equation
(
PDE
) is an equation which involves a
multivariable function
and one or more of its
partial derivatives
.
The function is often thought of as an "unknown" that solves the equation. However, it is often impossible to write down explicit formulas for solutions of partial differential equations. Hence there is a vast amount of modern mathematical and scientific research on methods to
numerically approximate
solutions of partial differential equations using computers. Partial differential equations also occupy a large sector of
pure mathematical research
, where the focus is on the qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability.
[
1
]
Among the many open questions are the
existence and smoothness
of solutions to the
Navier–Stokes equations
, named as one of the
Millennium Prize Problems
in 2000.
Partial differential equations occur very widely in mathematically oriented scientific fields, such as
physics
and
engineering
. For instance, they are foundational in the modern scientific understanding of
sound
,
heat
,
diffusion
,
electrostatics
,
electrodynamics
,
thermodynamics
,
fluid dynamics
,
elasticity
,
general relativity
, and
quantum mechanics
(
Schrödinger equation
,
Pauli equation
etc.). They also arise from many purely mathematical considerations, such as
differential geometry
and the
calculus of variations
; among other notable applications, they are the fundamental tool in the proof of the
Poincaré conjecture
from
geometric topology
.
Partly due to this variety of sources, there is a wide spectrum of types of partial differential equations. Many different methods have been developed for dealing with the individual equations which arise. As such, there is no "universal theory" of partial differential equations, with specialist knowledge being divided between several distinct subfields.
[
2
]
Ordinary differential equations
can be viewed as a subclass of partial differential equations, corresponding to
functions of a single variable
.
Stochastic partial differential equations
and
nonlocal equations
are widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include
elliptic
and
parabolic
partial differential equations,
fluid mechanics
,
Boltzmann equations
, and
dispersive partial differential equations
.
[
3
]
Introduction and examples
[
edit
]
One of the most important partial differential equations, with many applications, is
Laplace's equation
.
For a function
u
(
x
,
y
,
z
)
of three variables, Laplace's equation is
A function that obeys this equation is called a
harmonic function
.
Such functions were widely studied in the 19th century due to their relevance for
classical mechanics
. For example, the equilibrium temperature distribution of a homogeneous solid is a harmonic function. It is usually a matter of straightforward computation to check whether or not a given function is harmonic. For instance
and
are all harmonic, while
is not. It may be surprising that these examples of harmonic functions are of such different forms. This is a reflection of the fact that they are not special cases of a "general solution formula" of Laplace's equation. This is in striking contrast to the case of many
ordinary differential equations
(ODEs), where many introductory textbooks aim to find methods leading to general solutions. For Laplace's equation, as for a large number of partial differential equations, such solution formulas do not exist.
This can also be seen in the case of the following PDE: for a function
v
(
x
,
y
)
of two variables, consider the equation
It can be directly checked that any function
v
of the form
v
(
x
,
y
) =
f
(
x
) +
g
(
y
)
, for any single-variable (differentiable) functions
f
and
g
whatsoever, satisfies this condition. This is far beyond the choices available in ODE solution formulas, which typically allow the free choice of some numbers. In the study of PDEs, one generally has the free choice of functions.
The nature of this choice varies from PDE to PDE. To understand it for any given equation, existence and uniqueness theorems are usually important organizational principles. In many introductory textbooks, the role of
existence and uniqueness theorems for ODE
can be somewhat opaque; the existence half is usually unnecessary, since one can directly check any proposed solution formula, while the uniqueness half is often only present in the background to ensure that a proposed solution formula is as general as possible. By contrast, for PDE, existence and uniqueness theorems are often the only means by which one can navigate through the plethora of different solutions at hand. For this reason, they are also fundamental when carrying out a purely numerical simulation, as one must have an understanding of what data is to be prescribed by the user and what is to be left to the computer to calculate.
To discuss such existence and uniqueness theorems, it is necessary to be precise about the
domain
of the "unknown function". Otherwise, speaking only in terms such as "a function of two variables", it is impossible to meaningfully formulate the results. That is, the domain of the unknown function must be regarded as part of the structure of the PDE itself.
The following provides two classic examples of such existence and uniqueness theorems. Even though the two PDEs in question are so similar, there is a striking difference in behavior: for the first PDE, one has the free prescription of a single function, while for the second PDE, one has the free prescription of two functions.
Even more phenomena are possible. For instance, the
following PDE
, arising naturally in the field of
differential geometry
, illustrates an example where there is a simple and completely explicit solution formula, but with the free choice of only three numbers and not even one function.
If
u
is a function on
R
2
with
then there are numbers
a
,
b
, and
c
with
u
(
x
,
y
) =
ax
+
by
+
c
.
In contrast to the earlier examples, this PDE is
nonlinear
, owing to the square roots and the squares. A
linear
PDE is one such that, if it is homogeneous, the sum of any two solutions is also a solution, and any constant multiple of any solution is also a solution.
A partial differential equation is an equation that involves an unknown function of
variables and (some of) its partial derivatives.
[
4
]
That is, for the unknown function
of variables
belonging to the open subset
of
, the
-order partial differential equation is defined as
where
and
is the partial derivative operator.
When writing PDEs, it is common to denote partial derivatives using subscripts. For example:
In the general situation that
u
is a function of
n
variables, then
u
i
denotes the first partial derivative relative to the
i
-th input,
u
ij
denotes the second partial derivative relative to the
i
-th and
j
-th inputs, and so on.
The Greek letter
Δ
denotes the
Laplace operator
; if
u
is a function of
n
variables, then
In the physics literature, the Laplace operator is often denoted by
∇
2
; in the mathematics literature,
∇
2
u
may also denote the
Hessian matrix
of
u
.
Linear and nonlinear equations
[
edit
]
A PDE is called
linear
if it is linear in the unknown and its derivatives. For example, for a function
u
of
x
and
y
, a second order linear PDE is of the form
where
a
i
and
f
are functions of the independent variables
x
and
y
only. (Often the mixed-partial derivatives
u
xy
and
u
yx
will be equated, but this is not required for the discussion of linearity.)
If the
a
i
are constants (independent of
x
and
y
) then the PDE is called
linear with constant coefficients
. If
f
is zero everywhere then the linear PDE is
homogeneous
, otherwise it is
inhomogeneous
. (This is separate from
asymptotic homogenization
, which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.)
Nearest to linear PDEs are
semi-linear
PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order derivatives and the unknown function may appear arbitrarily. For example, a general second order semi-linear PDE in two variables is
In a
quasilinear
PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives:
Many of the fundamental PDEs in physics are quasilinear, such as the
Einstein equations
of
general relativity
and the
Navier–Stokes equations
describing fluid motion.
A PDE without any linearity properties is called
fully
nonlinear
, and possesses nonlinearities on one or more of the highest-order derivatives. An example is the
Monge–Ampère equation
, which arises in
differential geometry
.
[
5
]
Second order equations
[
edit
]
The elliptic/parabolic/hyperbolic classification provides a guide to appropriate
initial-
and
boundary conditions
and to the
smoothness
of the solutions. Assuming
u
xy
=
u
yx
, the general linear second-order PDE in two independent variables has the form
where the coefficients
A
,
B
,
C
... may depend upon
x
and
y
. If
A
2
+
B
2
+
C
2
> 0
over a region of the
xy
-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:
More precisely, replacing
∂
x
by
X
, and likewise for other variables (formally this is done by a
Fourier transform
), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a
homogeneous polynomial
, here a
quadratic form
) being most significant for the classification.
Just as one classifies
conic sections
and quadratic forms into parabolic, hyperbolic, and elliptic based on the
discriminant
B
2
− 4
AC
, the same can be done for a second-order PDE at a given point. However, the
discriminant
in a PDE is given by
B
2
−
AC
due to the convention of the
xy
term being
2
B
rather than
B
; formally, the discriminant (of the associated quadratic form) is
(2
B
)
2
− 4
AC
= 4(
B
2
−
AC
)
, with the factor of 4 dropped for simplicity.
B
2
−
AC
< 0
(
elliptic partial differential equation
): Solutions of
elliptic PDEs
are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of
Laplace's equation
are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where
x
< 0
. By a change of variables, the equation can always be expressed in the form:
where x and y correspond to changed variables. This justifies
Laplace equation
as an example of this type.
[
6
]
B
2
−
AC
= 0
(
parabolic partial differential equation
): Equations that are
parabolic
at every point can be transformed into a form analogous to the
heat equation
by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where
x
= 0
. By change of variables, the equation can always be expressed in the form:
where x correspond to changed variables. This justifies the
heat equation
, which is of the form
, as an example of this type.
[
6
]
B
2
−
AC
> 0
(
hyperbolic partial differential equation
):
hyperbolic
equations retain any discontinuities of functions or derivatives in the initial data. An example is the
wave equation
. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where
x
> 0
. By change of variables, the equation can always be expressed in the form:
where x and y correspond to changed variables. This justifies the
wave equation
as an example of this type.
[
6
]
If there are
n
independent variables
x
1
,
x
2
, …,
x
n
, a general linear partial differential equation of second order has the form
The classification depends upon the signature of the
eigenvalues
of the coefficient matrix
a
i
,
j
.
Elliptic: the eigenvalues are all positive or all negative.
Parabolic: the eigenvalues are all positive or all negative, except one that is zero.
Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues.
[
7
]
The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the
Laplace equation
, the
heat equation
, and the
wave equation
.
However, the classification only depends on linearity of the second-order terms and is therefore applicable to semi- and quasilinear PDEs as well. The basic types also extend to hybrids such as the
Euler–Tricomi equation
; varying from elliptic to hyperbolic for different
regions
of the domain, as well as higher-order PDEs, but such knowledge is more specialized.
Systems of first-order equations and characteristic surfaces
[
edit
]
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown
u
is now a
vector
with
m
components, and the coefficient matrices
A
ν
are
m
by
m
matrices for
ν
= 1, 2, …,
n
. The partial differential equation takes the form
where the coefficient matrices
A
ν
and the vector
B
may depend upon
x
and
u
. If a
hypersurface
S
is given in the implicit form
where
φ
has a non-zero gradient, then
S
is a
characteristic surface
for the
operator
L
at a given point if the characteristic form vanishes:
The geometric interpretation of this condition is as follows: if data for
u
are prescribed on the surface
S
, then it may be possible to determine the normal derivative of
u
on
S
from the differential equation. If the data on
S
and the differential equation determine the normal derivative of
u
on
S
, then
S
is non-characteristic. If the data on
S
and the differential equation
do not
determine the normal derivative of
u
on
S
, then the surface is
characteristic
, and the differential equation restricts the data on
S
: the differential equation is
internal
to
S
.
A first-order system
Lu
= 0
is
elliptic
if no surface is characteristic for
L
: the values of
u
on
S
and the differential equation always determine the normal derivative of
u
on
S
.
A first-order system is
hyperbolic
at a point if there is a
spacelike
surface
S
with normal
ξ
at that point. This means that, given any non-trivial vector
η
orthogonal to
ξ
, and a scalar multiplier
λ
, the equation
Q
(
λξ
+
η
) = 0
has
m
real roots
λ
1
,
λ
2
, …,
λ
m
. The system is
strictly hyperbolic
if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form
Q
(
ζ
) = 0
defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has
nm
sheets, and the axis
ζ
=
λξ
runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.
Analytical solutions
[
edit
]
Separation of variables
[
edit
]
Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a feature of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is
the
solution (this also applies to ODEs). We assume as an
ansatz
that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.
[
8
]
In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.
This is possible for simple PDEs, which are called
separable partial differential equations
, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to
diagonal matrices
– thinking of "the value for fixed
x
" as a coordinate, each coordinate can be understood separately.
This generalizes to the
method of characteristics
, and is also used in
integral transforms
.
Method of characteristics
[
edit
]
The characteristic surface in
n
=
2
-
dimensional space is called a
characteristic curve
.
[
9
]
In special cases, one can find characteristic curves on which the first-order PDE reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the
method of characteristics
.
More generally, applying the method to first-order PDEs in higher dimensions, one may find characteristic surfaces.
An
integral transform
may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.
An important example of this is
Fourier analysis
, which diagonalizes the heat equation using the
eigenbasis
of sinusoidal waves.
If the domain is finite or periodic, an infinite sum of solutions such as a
Fourier series
is appropriate, but an integral of solutions such as a
Fourier integral
is generally required for infinite domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral.
Change of variables
[
edit
]
Often a PDE can be reduced to a simpler form with a known solution by a suitable
change of variables
. For example, the
Black–Scholes equation
is reducible to the
heat equation
by the change of variables
[
10
]
Fundamental solution
[
edit
]
Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding the
fundamental solution
(the solution for a point source
), then taking the
convolution
with the boundary conditions to get the solution.
This is analogous in
signal processing
to understanding a filter by its
impulse response
.
Superposition principle
[
edit
]
The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example
sin
x
+ sin
x
= 2 sin
x
. The same principle can be observed in PDEs where the solutions may be real or complex and additive. If
u
1
and
u
2
are solutions of linear PDE in some function space
R
, then
u
=
c
1
u
1
+
c
2
u
2
with any constants
c
1
and
c
2
are also a solution of that PDE in the same function space.
Methods for non-linear equations
[
edit
]
There are no generally applicable analytical methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the
Cauchy–Kowalevski theorem
) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of
analysis
).
Nevertheless, some techniques can be used for several types of equations. The
h
-principle
is the most powerful method to solve
underdetermined
equations. The
Riquier–Janet theory
is an effective method for obtaining information about many analytic
overdetermined
systems.
The
method of characteristics
can be used in some very special cases to solve nonlinear partial differential equations.
[
11
]
In some cases, a PDE can be solved via
perturbation analysis
in which the solution is considered to be a correction to an equation with a known solution. Alternatives are
numerical analysis
techniques from simple
finite difference
schemes to the more mature
multigrid
and
finite element methods
. Many interesting problems in science and engineering are solved in this way using
computers
, sometimes high performance
supercomputers
.
From 1870
Sophus Lie
's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called
Lie groups
, be referred, to a common source; and that ordinary differential equations which admit the same
infinitesimal transformations
present comparable difficulties of integration. He also emphasized the subject of
transformations of contact
.
A general approach to solving PDEs uses the symmetry property of differential equations, the continuous
infinitesimal transformations
of solutions to solutions (
Lie theory
). Continuous
group theory
,
Lie algebras
and
differential geometry
are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its
Lax pairs
, recursion operators,
Bäcklund transform
and finally finding exact analytic solutions to the PDE.
Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.
Semi-analytical methods
[
edit
]
The
Adomian decomposition method
,
[
12
]
the
Lyapunov
artificial small parameter method, and his
homotopy perturbation method
are all special cases of the more general
homotopy analysis method
.
[
13
]
These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known
perturbation theory
, thus giving these methods greater flexibility and solution generality.
Numerical solutions
[
edit
]
The three most widely used
numerical methods to solve PDEs
are the
finite element method
(FEM),
finite volume methods
(FVM) and
finite difference methods
(FDM), as well other kind of methods called
meshfree methods
, which were made to solve problems where the aforementioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version
hp-FEM
. Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM),
extended finite element method
(XFEM),
spectral finite element method
(SFEM),
meshfree finite element method
,
discontinuous Galerkin finite element method
(DGFEM),
element-free Galerkin method
(EFGM),
interpolating element-free Galerkin method
(IEFGM), etc.
Finite element method
[
edit
]
The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for approximating solutions of partial differential equations (PDE) as well as of integral equations using a finite set of functions.
[
14
]
[
15
]
The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.
Finite difference method
[
edit
]
Finite-difference methods are numerical methods for approximating the solutions to differential equations using
finite difference
equations to approximate derivatives.
Finite volume method
[
edit
]
Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the
divergence theorem
. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design.
Physics informed neural networks have been used to solve partial differential equations in both forward and inverse problems in a data driven manner.
[
16
]
One example is the reconstructing fluid flow governed by the
Navier-Stokes equations
. Using physics informed neural networks does not require the often expensive mesh generation that conventional
CFD
methods rely on.
[
17
]
[
18
]
It is evident that geometric and physical constraints have a synergistic effect on neural PDE surrogates, thereby enhancing their efficacy in predicting stable and super long rollouts.
[
19
]
Weak solutions are functions that satisfy the PDE, yet in other meanings than regular sense. The meaning for this term may differ with context, and one of the most commonly used definitions is based on the notion of
distributions
.
An example
[
20
]
for the definition of a weak solution is as follows:
Consider the boundary-value problem given by:
where
denotes a second-order partial differential operator in
divergence form
.
We say a
is a weak solution if
for every
, which can be derived by a formal integral by parts.
An example for a weak solution is as follows:
is a weak solution satisfying
in distributional sense, as formally,
Theoretical studies
[
edit
]
In
pure mathematics
, the theoretical studies of PDEs focus on the criteria for a solution to exist and the properties of a solution while finding its formula is often secondary.
Well-posedness refers to a common schematic package of information about a PDE. To say that a PDE is well-posed, one must have:
an existence and uniqueness theorem, asserting that by the prescription of some freely chosen functions, one can single out one specific solution of the PDE
by
continuously
changing the free choices, one continuously changes the corresponding solution
This is, by the necessity of being applicable to several different PDE, somewhat vague. The requirement of "continuity", in particular, is ambiguous, since there are usually many inequivalent means by which it can be rigorously defined. It is, however, somewhat unusual to study a PDE without specifying a way in which it is well-posed.
Regularity refers to the integrability and differentiability of weak solutions, which can often be represented by
Sobolev spaces
.
This problem arise due to the difficulty in searching for classical solutions. Researchers often tend to find weak solutions at first and then find out whether it is smooth enough to be qualified as a classical solution.
Results from
functional analysis
are often used in this field of study.
Some common PDEs
Acoustic wave equation
Burgers' equation
Continuity equation
Heat equation
Helmholtz equation
Klein–Gordon equation
Jacobi equation
Lagrange equation
Lorenz equation
Laplace's equation
Maxwell's equations
Navier-Stokes equation
Poisson's equation
Reaction–diffusion system
Schrödinger equation
Wave equation
Types of boundary conditions
Dirichlet boundary condition
Neumann boundary condition
Robin boundary condition
Cauchy problem
Various topics
Jet bundle
Laplace transform applied to differential equations
List of dynamical systems and differential equations topics
Matrix differential equation
Numerical partial differential equations
Partial differential algebraic equation
Recurrence relation
Stochastic processes and boundary value problems
^
"Regularity and singularities in elliptic PDE's: beyond monotonicity formulas | EllipticPDE Project | Fact Sheet | H2020"
.
CORDIS | European Commission
. Retrieved
2024-02-05
.
^
Klainerman, Sergiu
(2010). "PDE as a Unified Subject". In Alon, N.; Bourgain, J.; Connes, A.; Gromov, M.; Milman, V. (eds.).
Visions in Mathematics
. Modern Birkhäuser Classics. Basel: Birkhäuser. pp.
279–
315.
doi
:
10.1007/978-3-0346-0422-2_10
.
ISBN
978-3-0346-0421-5
.
^
Erdoğan, M. Burak; Tzirakis, Nikolaos (2016).
Dispersive Partial Differential Equations: Wellposedness and Applications
. London Mathematical Society Student Texts. Cambridge: Cambridge University Press.
ISBN
978-1-107-14904-5
.
^
Evans 1998
, pp. 1–2.
^
Klainerman, Sergiu (2008), "Partial Differential Equations", in Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.),
The Princeton Companion to Mathematics
, Princeton University Press, pp.
455–
483
^
a
b
c
Levandosky, Julie.
"Classification of Second-Order Equations"
(PDF)
.
^
Courant and Hilbert (1962), p.182.
^
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Archived
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Solutions faibles des équations différentielles
at hal.science |
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## Contents
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- [1 Introduction and examples](https://en.wikipedia.org/wiki/Partial_differential_equation#Introduction_and_examples)
- [2 Definition](https://en.wikipedia.org/wiki/Partial_differential_equation#Definition)
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- [2\.1 Notation](https://en.wikipedia.org/wiki/Partial_differential_equation#Notation)
- [3 Classification](https://en.wikipedia.org/wiki/Partial_differential_equation#Classification)
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- [3\.1 Linear and nonlinear equations](https://en.wikipedia.org/wiki/Partial_differential_equation#Linear_and_nonlinear_equations)
- [3\.2 Second order equations](https://en.wikipedia.org/wiki/Partial_differential_equation#Second_order_equations)
- [3\.3 Systems of first-order equations and characteristic surfaces](https://en.wikipedia.org/wiki/Partial_differential_equation#Systems_of_first-order_equations_and_characteristic_surfaces)
- [4 Analytical solutions](https://en.wikipedia.org/wiki/Partial_differential_equation#Analytical_solutions)
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- [4\.1 Separation of variables](https://en.wikipedia.org/wiki/Partial_differential_equation#Separation_of_variables)
- [4\.2 Method of characteristics](https://en.wikipedia.org/wiki/Partial_differential_equation#Method_of_characteristics)
- [4\.3 Integral transform](https://en.wikipedia.org/wiki/Partial_differential_equation#Integral_transform)
- [4\.4 Change of variables](https://en.wikipedia.org/wiki/Partial_differential_equation#Change_of_variables)
- [4\.5 Fundamental solution](https://en.wikipedia.org/wiki/Partial_differential_equation#Fundamental_solution)
- [4\.6 Superposition principle](https://en.wikipedia.org/wiki/Partial_differential_equation#Superposition_principle)
- [4\.7 Methods for non-linear equations](https://en.wikipedia.org/wiki/Partial_differential_equation#Methods_for_non-linear_equations)
- [4\.8 Lie group method](https://en.wikipedia.org/wiki/Partial_differential_equation#Lie_group_method)
- [4\.9 Semi-analytical methods](https://en.wikipedia.org/wiki/Partial_differential_equation#Semi-analytical_methods)
- [5 Numerical solutions](https://en.wikipedia.org/wiki/Partial_differential_equation#Numerical_solutions)
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- [5\.1 Finite element method](https://en.wikipedia.org/wiki/Partial_differential_equation#Finite_element_method)
- [5\.2 Finite difference method](https://en.wikipedia.org/wiki/Partial_differential_equation#Finite_difference_method)
- [5\.3 Finite volume method](https://en.wikipedia.org/wiki/Partial_differential_equation#Finite_volume_method)
- [5\.4 Neural networks](https://en.wikipedia.org/wiki/Partial_differential_equation#Neural_networks)
- [6 Weak solutions](https://en.wikipedia.org/wiki/Partial_differential_equation#Weak_solutions)
- [7 Theoretical studies](https://en.wikipedia.org/wiki/Partial_differential_equation#Theoretical_studies)
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- [7\.1 Well-posedness](https://en.wikipedia.org/wiki/Partial_differential_equation#Well-posedness)
- [7\.2 Regularity](https://en.wikipedia.org/wiki/Partial_differential_equation#Regularity)
- [8 See also](https://en.wikipedia.org/wiki/Partial_differential_equation#See_also)
- [9 Notes](https://en.wikipedia.org/wiki/Partial_differential_equation#Notes)
- [10 References](https://en.wikipedia.org/wiki/Partial_differential_equation#References)
- [11 Further reading](https://en.wikipedia.org/wiki/Partial_differential_equation#Further_reading)
- [12 External links](https://en.wikipedia.org/wiki/Partial_differential_equation#External_links)
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# Partial differential equation
50 languages
- [العربية](https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A9_%D8%AA%D9%81%D8%A7%D8%B6%D9%84%D9%8A%D8%A9_%D8%AC%D8%B2%D8%A6%D9%8A%D8%A9 "معادلة تفاضلية جزئية – Arabic")
- [الدارجة](https://ary.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A9_%D8%AA%D9%81%D8%A7%D8%B6%D9%84%D9%8A%D8%A9_%D8%AC%D8%B2%D8%A6%D9%8A%D8%A9 "معادلة تفاضلية جزئية – Moroccan Arabic")
- [Asturianu](https://ast.wikipedia.org/wiki/Ecuaci%C3%B3n_en_derivaes_parciales "Ecuación en derivaes parciales – Asturian")
- [Български](https://bg.wikipedia.org/wiki/%D0%A7%D0%B0%D1%81%D1%82%D0%BD%D0%BE_%D0%B4%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%BE_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5 "Частно диференциално уравнение – Bulgarian")
- [বাংলা](https://bn.wikipedia.org/wiki/%E0%A6%86%E0%A6%82%E0%A6%B6%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A7%8D%E0%A6%AF%E0%A6%AC%E0%A6%95%E0%A6%B2%E0%A6%A8%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3 "আংশিক ব্যবকলনীয় সমীকরণ – Bangla")
- [Català](https://ca.wikipedia.org/wiki/Equaci%C3%B3_diferencial_en_derivades_parcials "Equació diferencial en derivades parcials – Catalan")
- [Čeština](https://cs.wikipedia.org/wiki/Parci%C3%A1ln%C3%AD_diferenci%C3%A1ln%C3%AD_rovnice "Parciální diferenciální rovnice – Czech")
- [Чӑвашла](https://cv.wikipedia.org/wiki/%D0%A5%D0%B0%D1%80%D0%BF%C4%83%D1%80_%D1%82%C4%83%D1%85%C4%83%D0%BC%D1%81%D0%B5%D0%BC%D0%BB%C4%95_%D0%B4%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BB%C4%83_%D1%82%D0%B0%D0%BD%D0%BB%C4%83%D1%85 "Харпăр тăхăмсемлĕ дифференциаллă танлăх – Chuvash")
- [Deutsch](https://de.wikipedia.org/wiki/Partielle_Differentialgleichung "Partielle Differentialgleichung – German")
- [Ελληνικά](https://el.wikipedia.org/wiki/%CE%9C%CE%B5%CF%81%CE%B9%CE%BA%CE%AE_%CE%B4%CE%B9%CE%B1%CF%86%CE%BF%CF%81%CE%B9%CE%BA%CE%AE_%CE%B5%CE%BE%CE%AF%CF%83%CF%89%CF%83%CE%B7 "Μερική διαφορική εξίσωση – Greek")
- [Esperanto](https://eo.wikipedia.org/wiki/Parta_diferenciala_ekvacio "Parta diferenciala ekvacio – Esperanto")
- [Español](https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_en_derivadas_parciales "Ecuación en derivadas parciales – Spanish")
- [Eesti](https://et.wikipedia.org/wiki/Osatuletistega_diferentsiaalv%C3%B5rrand "Osatuletistega diferentsiaalvõrrand – Estonian")
- [Euskara](https://eu.wikipedia.org/wiki/Ekuazio_diferentzial_partzial "Ekuazio diferentzial partzial – Basque")
- [فارسی](https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D9%87_%D8%AF%DB%8C%D9%81%D8%B1%D8%A7%D9%86%D8%B3%DB%8C%D9%84_%D8%A8%D8%A7_%D9%85%D8%B4%D8%AA%D9%82%D8%A7%D8%AA_%D8%AC%D8%B2%D8%A6%DB%8C "معادله دیفرانسیل با مشتقات جزئی – Persian")
- [Suomi](https://fi.wikipedia.org/wiki/Osittaisdifferentiaaliyht%C3%A4l%C3%B6 "Osittaisdifferentiaaliyhtälö – Finnish")
- [Français](https://fr.wikipedia.org/wiki/%C3%89quation_aux_d%C3%A9riv%C3%A9es_partielles "Équation aux dérivées partielles – French")
- [Galego](https://gl.wikipedia.org/wiki/Ecuaci%C3%B3n_en_derivadas_parciais "Ecuación en derivadas parciais – Galician")
- [עברית](https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%94_%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA_%D7%97%D7%9C%D7%A7%D7%99%D7%AA "משוואה דיפרנציאלית חלקית – Hebrew")
- [हिन्दी](https://hi.wikipedia.org/wiki/%E0%A4%86%E0%A4%82%E0%A4%B6%E0%A4%BF%E0%A4%95_%E0%A4%85%E0%A4%B5%E0%A4%95%E0%A4%B2_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3 "आंशिक अवकल समीकरण – Hindi")
- [Magyar](https://hu.wikipedia.org/wiki/Parci%C3%A1lis_differenci%C3%A1legyenlet "Parciális differenciálegyenlet – Hungarian")
- [Հայերեն](https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%BD%D5%B6%D5%A1%D5%AF%D5%AB_%D5%A1%D5%AE%D5%A1%D5%B6%D6%81%D5%B5%D5%A1%D5%AC%D5%B6%D5%A5%D6%80%D5%B8%D5%BE_%D5%A4%D5%AB%D6%86%D5%A5%D6%80%D5%A5%D5%B6%D6%81%D5%AB%D5%A1%D5%AC_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4%D5%B6%D5%A5%D6%80 "Մասնակի ածանցյալներով դիֆերենցիալ հավասարումներ – Armenian")
- [Bahasa Indonesia](https://id.wikipedia.org/wiki/Persamaan_diferensial_parsial "Persamaan diferensial parsial – Indonesian")
- [Italiano](https://it.wikipedia.org/wiki/Equazione_differenziale_alle_derivate_parziali "Equazione differenziale alle derivate parziali – Italian")
- [日本語](https://ja.wikipedia.org/wiki/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%BC%8F "偏微分方程式 – Japanese")
- [한국어](https://ko.wikipedia.org/wiki/%ED%8E%B8%EB%AF%B8%EB%B6%84_%EB%B0%A9%EC%A0%95%EC%8B%9D "편미분 방정식 – Korean")
- [Македонски](https://mk.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B4%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B0 "Парцијална диференцијална равенка – Macedonian")
- [Bahasa Melayu](https://ms.wikipedia.org/wiki/Persamaan_pembezaan_separa "Persamaan pembezaan separa – Malay")
- [Nederlands](https://nl.wikipedia.org/wiki/Parti%C3%ABle_differentiaalvergelijking "Partiële differentiaalvergelijking – Dutch")
- [Norsk bokmål](https://no.wikipedia.org/wiki/Partielle_differensialligninger "Partielle differensialligninger – Norwegian Bokmål")
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- [Português](https://pt.wikipedia.org/wiki/Equa%C3%A7%C3%A3o_diferencial_parcial "Equação diferencial parcial – Portuguese")
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- [Русский](https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B2_%D1%87%D0%B0%D1%81%D1%82%D0%BD%D1%8B%D1%85_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D0%BD%D1%8B%D1%85 "Дифференциальное уравнение в частных производных – Russian")
- [Scots](https://sco.wikipedia.org/wiki/Pairtial_differential_equation "Pairtial differential equation – Scots")
- [Srpskohrvatski / српскохрватски](https://sh.wikipedia.org/wiki/Parcijalna_diferencijalna_jedna%C4%8Dina "Parcijalna diferencijalna jednačina – Serbo-Croatian")
- [Simple English](https://simple.wikipedia.org/wiki/Partial_differential_equation "Partial differential equation – Simple English")
- [Slovenčina](https://sk.wikipedia.org/wiki/Parci%C3%A1lna_diferenci%C3%A1lna_rovnica "Parciálna diferenciálna rovnica – Slovak")
- [Slovenščina](https://sl.wikipedia.org/wiki/Parcialna_diferencialna_ena%C4%8Dba "Parcialna diferencialna enačba – Slovenian")
- [Shqip](https://sq.wikipedia.org/wiki/Ekuacionet_diferenciale_t%C3%AB_pjesshme "Ekuacionet diferenciale të pjesshme – Albanian")
- [Српски / srpski](https://sr.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B4%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D1%98%D0%B5%D0%B4%D0%BD%D0%B0%D1%87%D0%B8%D0%BD%D0%B0 "Парцијална диференцијална једначина – Serbian")
- [Svenska](https://sv.wikipedia.org/wiki/Partiell_differentialekvation "Partiell differentialekvation – Swedish")
- [ไทย](https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%A1%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%AD%E0%B8%99%E0%B8%B8%E0%B8%9E%E0%B8%B1%E0%B8%99%E0%B8%98%E0%B9%8C%E0%B8%A2%E0%B9%88%E0%B8%AD%E0%B8%A2 "สมการเชิงอนุพันธ์ย่อย – Thai")
- [Tagalog](https://tl.wikipedia.org/wiki/Ekwasyong_parsiyal_diperensiyal "Ekwasyong parsiyal diperensiyal – Tagalog")
- [Türkçe](https://tr.wikipedia.org/wiki/K%C4%B1smi_diferansiyel_denklem "Kısmi diferansiyel denklem – Turkish")
- [Українська](https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B5_%D1%80%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F_%D0%B7_%D1%87%D0%B0%D1%81%D1%82%D0%B8%D0%BD%D0%BD%D0%B8%D0%BC%D0%B8_%D0%BF%D0%BE%D1%85%D1%96%D0%B4%D0%BD%D0%B8%D0%BC%D0%B8 "Диференціальне рівняння з частинними похідними – Ukrainian")
- [Oʻzbekcha / ўзбекча](https://uz.wikipedia.org/wiki/Xususiy_hosilali_differensial_tenglama "Xususiy hosilali differensial tenglama – Uzbek")
- [Tiếng Việt](https://vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_vi_ph%C3%A2n_ri%C3%AAng_ph%E1%BA%A7n "Phương trình vi phân riêng phần – Vietnamese")
- [粵語](https://zh-yue.wikipedia.org/wiki/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B "偏微分方程 – Cantonese")
- [中文](https://zh.wikipedia.org/wiki/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B "偏微分方程 – Chinese")
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From Wikipedia, the free encyclopedia
Type of differential equation
| | |
|---|---|
|  | This article includes a list of [general references](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources#General_references "Wikipedia:Citing sources"), but **it lacks sufficient corresponding [inline citations](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources#Inline_citations "Wikipedia:Citing sources")**. Please help to [improve](https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Reliability "Wikipedia:WikiProject Reliability") this article by [introducing](https://en.wikipedia.org/wiki/Wikipedia:When_to_cite "Wikipedia:When to cite") more precise citations. *(March 2023)* *([Learn how and when to remove this message](https://en.wikipedia.org/wiki/Help:Maintenance_template_removal "Help:Maintenance template removal"))* |
| [Differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation") |
|---|
| Scope |
| [Natural sciences](https://en.wikipedia.org/wiki/Natural_science "Natural science") [Engineering](https://en.wikipedia.org/wiki/Engineering "Engineering") |
| [Astronomy](https://en.wikipedia.org/wiki/Astronomy "Astronomy") [Physics](https://en.wikipedia.org/wiki/Physics "Physics") [Chemistry](https://en.wikipedia.org/wiki/Chemistry "Chemistry") [Biology](https://en.wikipedia.org/wiki/Biology "Biology") [Geology](https://en.wikipedia.org/wiki/Geology "Geology") |
| [Applied mathematics](https://en.wikipedia.org/wiki/Applied_mathematics "Applied mathematics") |
| [Continuum mechanics](https://en.wikipedia.org/wiki/Continuum_mechanics "Continuum mechanics") [Chaos theory](https://en.wikipedia.org/wiki/Chaos_theory "Chaos theory") [Dynamical systems](https://en.wikipedia.org/wiki/Dynamical_systems "Dynamical systems") |
| [Social sciences](https://en.wikipedia.org/wiki/Social_science "Social science") |
| [Economics](https://en.wikipedia.org/wiki/Economics "Economics") [Population dynamics](https://en.wikipedia.org/wiki/Population_dynamics "Population dynamics") |
| Classification |
| |
| [Ordinary](https://en.wikipedia.org/wiki/Ordinary_differential_equation "Ordinary differential equation") [Partial]() [Differential-algebraic](https://en.wikipedia.org/wiki/Differential-algebraic_system_of_equations "Differential-algebraic system of equations") [Integro-differential](https://en.wikipedia.org/wiki/Integro-differential_equation "Integro-differential equation") [Fractional](https://en.wikipedia.org/wiki/Fractional_differential_equations "Fractional differential equations") [Linear](https://en.wikipedia.org/wiki/Linear_differential_equation "Linear differential equation") [Non-linear](https://en.wikipedia.org/wiki/Nonlinear_system#Nonlinear_differential_equations "Nonlinear system") |
| By variable type |
| [Dependent and independent variables](https://en.wikipedia.org/wiki/Dependent_and_independent_variables "Dependent and independent variables") [Autonomous](https://en.wikipedia.org/wiki/Autonomous_differential_equation "Autonomous differential equation") Coupled / Decoupled [Exact](https://en.wikipedia.org/wiki/Exact_differential_equation "Exact differential equation") [Homogeneous](https://en.wikipedia.org/wiki/Homogeneous_differential_equation "Homogeneous differential equation") / [Nonhomogeneous](https://en.wikipedia.org/wiki/Non-homogeneous_differential_equation "Non-homogeneous differential equation") |
| Features |
| [Order](https://en.wikipedia.org/wiki/Ordinary_differential_equation#Definitions "Ordinary differential equation") [Operator](https://en.wikipedia.org/wiki/Differential_operator "Differential operator") [Notation](https://en.wikipedia.org/wiki/Notation_for_differentiation "Notation for differentiation") |
| Relation to processes [Difference (discrete analogue)](https://en.wikipedia.org/wiki/Difference_equation "Difference equation") [Stochastic](https://en.wikipedia.org/wiki/Stochastic_differential_equation "Stochastic differential equation") [Stochastic partial](https://en.wikipedia.org/wiki/Stochastic_partial_differential_equation "Stochastic partial differential equation") [Delay](https://en.wikipedia.org/wiki/Delay_differential_equation "Delay differential equation") |
| Solution |
| Existence and uniqueness [Well-posed problem](https://en.wikipedia.org/wiki/Well-posed_problem "Well-posed problem") [Picard–Lindelöf theorem](https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem "Picard–Lindelöf theorem") [Peano existence theorem](https://en.wikipedia.org/wiki/Peano_existence_theorem "Peano existence theorem") [Carathéodory's existence theorem](https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_existence_theorem "Carathéodory's existence theorem") [Cauchy–Kovalevskaya theorem](https://en.wikipedia.org/wiki/Cauchy%E2%80%93Kovalevskaya_theorem "Cauchy–Kovalevskaya theorem") |
| General topics [Initial values](https://en.wikipedia.org/wiki/Initial_value_problem "Initial value problem") [Boundary values](https://en.wikipedia.org/wiki/Boundary_value_problem "Boundary value problem") [Dirichlet](https://en.wikipedia.org/wiki/Dirichlet_boundary_condition "Dirichlet boundary condition") [Neumann](https://en.wikipedia.org/wiki/Neumann_boundary_condition "Neumann boundary condition") [Robin](https://en.wikipedia.org/wiki/Robin_boundary_condition "Robin boundary condition") [Cauchy](https://en.wikipedia.org/wiki/Cauchy_problem "Cauchy problem") [Wronskian](https://en.wikipedia.org/wiki/Wronskian "Wronskian") [Abel's identity](https://en.wikipedia.org/wiki/Abel%27s_identity "Abel's identity") [Sturm–Liouville theory](https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory "Sturm–Liouville theory") [Phase portrait](https://en.wikipedia.org/wiki/Phase_portrait "Phase portrait") Stability [Lyapunov](https://en.wikipedia.org/wiki/Lyapunov_stability "Lyapunov stability") [Asymptotic](https://en.wikipedia.org/wiki/Asymptotic_stability "Asymptotic stability") [Exponential](https://en.wikipedia.org/wiki/Exponential_stability "Exponential stability") [Series solutions](https://en.wikipedia.org/wiki/Power_series_solution_of_differential_equations "Power series solution of differential equations") [Rate of convergence](https://en.wikipedia.org/wiki/Rate_of_convergence "Rate of convergence") [Asymptotic series](https://en.wikipedia.org/wiki/Asymptotic_expansion "Asymptotic expansion") [Special functions](https://en.wikipedia.org/wiki/Special_functions "Special functions") [Numerical integration](https://en.wikipedia.org/wiki/Numerical_integration "Numerical integration") [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function "Dirac delta function") |
| Solution methods Inspection [Method of characteristics](https://en.wikipedia.org/wiki/Method_of_characteristics "Method of characteristics") [Ansatz](https://en.wikipedia.org/wiki/Ansatz "Ansatz") [Euler](https://en.wikipedia.org/wiki/Euler_method "Euler method") [Exponential response formula](https://en.wikipedia.org/wiki/Exponential_response_formula "Exponential response formula") [Finite difference](https://en.wikipedia.org/wiki/Finite_difference_method "Finite difference method") [Crank–Nicolson](https://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method "Crank–Nicolson method") [Finite element](https://en.wikipedia.org/wiki/Finite_element_method "Finite element method") [Infinite element](https://en.wikipedia.org/wiki/Infinite_element_method "Infinite element method") [Finite volume](https://en.wikipedia.org/wiki/Finite_volume_method "Finite volume method") [Galerkin](https://en.wikipedia.org/wiki/Galerkin_method "Galerkin method") [Petrov–Galerkin](https://en.wikipedia.org/wiki/Petrov%E2%80%93Galerkin_method "Petrov–Galerkin method") [Green's function](https://en.wikipedia.org/wiki/Green%27s_function "Green's function") [Integrating factor](https://en.wikipedia.org/wiki/Integrating_factor "Integrating factor") [Integral transforms](https://en.wikipedia.org/wiki/Integral_transform "Integral transform") [Perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory "Perturbation theory") [Reduction of order](https://en.wikipedia.org/wiki/Reduction_of_order "Reduction of order") [Runge–Kutta](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods "Runge–Kutta methods") [Separation of variables](https://en.wikipedia.org/wiki/Separation_of_variables "Separation of variables") [Undetermined coefficients](https://en.wikipedia.org/wiki/Method_of_undetermined_coefficients "Method of undetermined coefficients") [Variation of parameters](https://en.wikipedia.org/wiki/Variation_of_parameters "Variation of parameters") [WKB approximation](https://en.wikipedia.org/wiki/WKB_approximation "WKB approximation") |
| People |
| List [Newton](https://en.wikipedia.org/wiki/Isaac_Newton "Isaac Newton") [Leibniz](https://en.wikipedia.org/wiki/Gottfried_Leibniz "Gottfried Leibniz") [Jacob Bernoulli](https://en.wikipedia.org/wiki/Jacob_Bernoulli "Jacob Bernoulli") [d'Alembert](https://en.wikipedia.org/wiki/Jean_Le_Rond_d%27Alembert "Jean Le Rond d'Alembert") [Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") [Lagrange](https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange "Joseph-Louis Lagrange") [Laplace](https://en.wikipedia.org/wiki/Pierre-Simon_Laplace "Pierre-Simon Laplace") [Wroński](https://en.wikipedia.org/wiki/J%C3%B3zef_Maria_Hoene-Wro%C5%84ski "Józef Maria Hoene-Wroński") [Fourier](https://en.wikipedia.org/wiki/Joseph_Fourier "Joseph Fourier") [Cauchy](https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy "Augustin-Louis Cauchy") [Green](https://en.wikipedia.org/wiki/George_Green_\(mathematician\) "George Green (mathematician)") [Dirichlet](https://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlet "Peter Gustav Lejeune Dirichlet") [Sturm](https://en.wikipedia.org/wiki/Jacques_Charles_Fran%C3%A7ois_Sturm "Jacques Charles François Sturm") [Liouville](https://en.wikipedia.org/wiki/Joseph_Liouville "Joseph Liouville") [Neumann](https://en.wikipedia.org/wiki/Carl_Neumann "Carl Neumann") [Robin](https://en.wikipedia.org/wiki/Victor_Gustave_Robin "Victor Gustave Robin") [Boole](https://en.wikipedia.org/wiki/George_Boole "George Boole") [Kovalevskaya](https://en.wikipedia.org/wiki/Sofya_Kovalevskaya "Sofya Kovalevskaya") [Runge](https://en.wikipedia.org/wiki/Carl_David_Tolm%C3%A9_Runge "Carl David Tolmé Runge") [Kutta](https://en.wikipedia.org/wiki/Martin_Kutta "Martin Kutta") [Lipschitz](https://en.wikipedia.org/wiki/Rudolf_Lipschitz "Rudolf Lipschitz") [Lindelöf](https://en.wikipedia.org/wiki/Ernst_Lindel%C3%B6f "Ernst Lindelöf") [Picard](https://en.wikipedia.org/wiki/%C3%89mile_Picard "Émile Picard") [Jeffreys](https://en.wikipedia.org/wiki/Harold_Jeffreys "Harold Jeffreys") [Nicolson](https://en.wikipedia.org/wiki/Phyllis_Nicolson "Phyllis Nicolson") [Crank](https://en.wikipedia.org/wiki/John_Crank "John Crank") |
| [Named equations](https://en.wikipedia.org/wiki/List_of_named_differential_equations "List of named differential equations") |
| [v](https://en.wikipedia.org/wiki/Template:Differential_equations "Template:Differential equations") [t](https://en.wikipedia.org/wiki/Template_talk:Differential_equations "Template talk:Differential equations") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Differential_equations "Special:EditPage/Template:Differential equations") |
[](https://en.wikipedia.org/wiki/File:Heat.gif)
A visualisation of a solution to the two-dimensional [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation") with temperature represented by the vertical direction and color
In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), a **partial differential equation** (**PDE**) is an equation which involves a [multivariable function](https://en.wikipedia.org/wiki/Function_of_several_real_variables "Function of several real variables") and one or more of its [partial derivatives](https://en.wikipedia.org/wiki/Partial_derivative "Partial derivative").
The function is often thought of as an "unknown" that solves the equation. However, it is often impossible to write down explicit formulas for solutions of partial differential equations. Hence there is a vast amount of modern mathematical and scientific research on methods to [numerically approximate](https://en.wikipedia.org/wiki/Numerical_methods_for_partial_differential_equations "Numerical methods for partial differential equations") solutions of partial differential equations using computers. Partial differential equations also occupy a large sector of [pure mathematical research](https://en.wikipedia.org/wiki/Pure_mathematics "Pure mathematics"), where the focus is on the qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability.[\[1\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-1) Among the many open questions are the [existence and smoothness](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness "Navier–Stokes existence and smoothness") of solutions to the [Navier–Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations "Navier–Stokes equations"), named as one of the [Millennium Prize Problems](https://en.wikipedia.org/wiki/Millennium_Prize_Problems "Millennium Prize Problems") in 2000.
Partial differential equations occur very widely in mathematically oriented scientific fields, such as [physics](https://en.wikipedia.org/wiki/Physics "Physics") and [engineering](https://en.wikipedia.org/wiki/Engineering "Engineering"). For instance, they are foundational in the modern scientific understanding of [sound](https://en.wikipedia.org/wiki/Sound "Sound"), [heat](https://en.wikipedia.org/wiki/Heat "Heat"), [diffusion](https://en.wikipedia.org/wiki/Diffusion "Diffusion"), [electrostatics](https://en.wikipedia.org/wiki/Electrostatics "Electrostatics"), [electrodynamics](https://en.wikipedia.org/wiki/Electromagnetism "Electromagnetism"), [thermodynamics](https://en.wikipedia.org/wiki/Thermodynamics "Thermodynamics"), [fluid dynamics](https://en.wikipedia.org/wiki/Fluid_dynamics "Fluid dynamics"), [elasticity](https://en.wikipedia.org/wiki/Elasticity_\(physics\) "Elasticity (physics)"), [general relativity](https://en.wikipedia.org/wiki/General_relativity "General relativity"), and [quantum mechanics](https://en.wikipedia.org/wiki/Quantum_mechanics "Quantum mechanics") ([Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation"), [Pauli equation](https://en.wikipedia.org/wiki/Pauli_equation "Pauli equation") etc.). They also arise from many purely mathematical considerations, such as [differential geometry](https://en.wikipedia.org/wiki/Differential_geometry "Differential geometry") and the [calculus of variations](https://en.wikipedia.org/wiki/Calculus_of_variations "Calculus of variations"); among other notable applications, they are the fundamental tool in the proof of the [Poincaré conjecture](https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture "Poincaré conjecture") from [geometric topology](https://en.wikipedia.org/wiki/Geometric_topology "Geometric topology").
Partly due to this variety of sources, there is a wide spectrum of types of partial differential equations. Many different methods have been developed for dealing with the individual equations which arise. As such, there is no "universal theory" of partial differential equations, with specialist knowledge being divided between several distinct subfields.[\[2\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-2)
[Ordinary differential equations](https://en.wikipedia.org/wiki/Ordinary_differential_equation "Ordinary differential equation") can be viewed as a subclass of partial differential equations, corresponding to [functions of a single variable](https://en.wikipedia.org/wiki/Function_of_a_real_variable "Function of a real variable"). [Stochastic partial differential equations](https://en.wikipedia.org/wiki/Stochastic_partial_differential_equation "Stochastic partial differential equation") and [nonlocal equations](https://en.wikipedia.org/wiki/Fractional_calculus "Fractional calculus") are widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include [elliptic](https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation "Elliptic partial differential equation") and [parabolic](https://en.wikipedia.org/wiki/Parabolic_partial_differential_equation "Parabolic partial differential equation") partial differential equations, [fluid mechanics](https://en.wikipedia.org/wiki/Fluid_mechanics "Fluid mechanics"), [Boltzmann equations](https://en.wikipedia.org/wiki/Boltzmann_equation "Boltzmann equation"), and [dispersive partial differential equations](https://en.wikipedia.org/wiki/Dispersive_partial_differential_equation "Dispersive partial differential equation").[\[3\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-3)
## Introduction and examples
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=1 "Edit section: Introduction and examples")\]
One of the most important partial differential equations, with many applications, is [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation"). For a function *u*(*x*, *y*, *z*) of three variables, Laplace's equation is ∂ 2 u ∂ x 2 \+ ∂ 2 u ∂ y 2 \+ ∂ 2 u ∂ z 2 \= 0\. {\\displaystyle {\\frac {\\partial ^{2}u}{\\partial x^{2}}}+{\\frac {\\partial ^{2}u}{\\partial y^{2}}}+{\\frac {\\partial ^{2}u}{\\partial z^{2}}}=0.}  A function that obeys this equation is called a [harmonic function](https://en.wikipedia.org/wiki/Harmonic_function "Harmonic function"). Such functions were widely studied in the 19th century due to their relevance for [classical mechanics](https://en.wikipedia.org/wiki/Classical_mechanics "Classical mechanics"). For example, the equilibrium temperature distribution of a homogeneous solid is a harmonic function. It is usually a matter of straightforward computation to check whether or not a given function is harmonic. For instance u ( x , y , z ) \= 1 x 2 − 2 x \+ y 2 \+ z 2 \+ 1 , {\\displaystyle u(x,y,z)={\\frac {1}{\\sqrt {x^{2}-2x+y^{2}+z^{2}+1}}},}  u ( x , y , z ) \= e 5 x sin ( 3 y ) cos ( 4 z ) {\\displaystyle u(x,y,z)=e^{5x}\\sin(3y)\\cos(4z)}  and u ( x , y , z ) \= 2 x 2 − y 2 − z 2 {\\displaystyle u(x,y,z)=2x^{2}-y^{2}-z^{2}}  are all harmonic, while u ( x , y , z ) \= sin ( x y ) \+ z {\\displaystyle u(x,y,z)=\\sin(xy)+z}  is not. It may be surprising that these examples of harmonic functions are of such different forms. This is a reflection of the fact that they are not special cases of a "general solution formula" of Laplace's equation. This is in striking contrast to the case of many [ordinary differential equations](https://en.wikipedia.org/wiki/Ordinary_differential_equation "Ordinary differential equation") (ODEs), where many introductory textbooks aim to find methods leading to general solutions. For Laplace's equation, as for a large number of partial differential equations, such solution formulas do not exist.
This can also be seen in the case of the following PDE: for a function *v*(*x*, *y*) of two variables, consider the equation ∂ 2 v ∂ x ∂ y \= 0\. {\\displaystyle {\\frac {\\partial ^{2}v}{\\partial x\\partial y}}=0.}  It can be directly checked that any function v of the form *v*(*x*, *y*) = *f*(*x*) + *g*(*y*), for any single-variable (differentiable) functions f and g whatsoever, satisfies this condition. This is far beyond the choices available in ODE solution formulas, which typically allow the free choice of some numbers. In the study of PDEs, one generally has the free choice of functions.
The nature of this choice varies from PDE to PDE. To understand it for any given equation, existence and uniqueness theorems are usually important organizational principles. In many introductory textbooks, the role of [existence and uniqueness theorems for ODE](https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem "Picard–Lindelöf theorem") can be somewhat opaque; the existence half is usually unnecessary, since one can directly check any proposed solution formula, while the uniqueness half is often only present in the background to ensure that a proposed solution formula is as general as possible. By contrast, for PDE, existence and uniqueness theorems are often the only means by which one can navigate through the plethora of different solutions at hand. For this reason, they are also fundamental when carrying out a purely numerical simulation, as one must have an understanding of what data is to be prescribed by the user and what is to be left to the computer to calculate.
To discuss such existence and uniqueness theorems, it is necessary to be precise about the [domain](https://en.wikipedia.org/wiki/Domain_of_a_function "Domain of a function") of the "unknown function". Otherwise, speaking only in terms such as "a function of two variables", it is impossible to meaningfully formulate the results. That is, the domain of the unknown function must be regarded as part of the structure of the PDE itself.
The following provides two classic examples of such existence and uniqueness theorems. Even though the two PDEs in question are so similar, there is a striking difference in behavior: for the first PDE, one has the free prescription of a single function, while for the second PDE, one has the free prescription of two functions.
- Let B denote the unit-radius disk around the origin in the plane. For any continuous function U on the unit circle, there is exactly one function u on B such that
∂
2
u
∂
x
2
\+
∂
2
u
∂
y
2
\=
0
{\\displaystyle {\\frac {\\partial ^{2}u}{\\partial x^{2}}}+{\\frac {\\partial ^{2}u}{\\partial y^{2}}}=0}
and whose restriction to the unit circle is given by U.
- For any functions f and g on the real line **R**, there is exactly one function u on **R** × (−1, 1) such that
∂
2
u
∂
x
2
−
∂
2
u
∂
y
2
\=
0
{\\displaystyle {\\frac {\\partial ^{2}u}{\\partial x^{2}}}-{\\frac {\\partial ^{2}u}{\\partial y^{2}}}=0}
and with *u*(*x*, 0) = *f*(*x*) and ∂*u*/∂*y*(*x*, 0) = *g*(*x*) for all values of x.
Even more phenomena are possible. For instance, the [following PDE](https://en.wikipedia.org/wiki/Bernstein%27s_problem "Bernstein's problem"), arising naturally in the field of [differential geometry](https://en.wikipedia.org/wiki/Differential_geometry "Differential geometry"), illustrates an example where there is a simple and completely explicit solution formula, but with the free choice of only three numbers and not even one function.
- If u is a function on **R**2 with
∂
∂
x
∂
u
∂
x
1
\+
(
∂
u
∂
x
)
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\+
(
∂
u
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)
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∂
∂
y
∂
u
∂
y
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(
∂
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∂
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{\\displaystyle {\\frac {\\partial }{\\partial x}}{\\frac {\\frac {\\partial u}{\\partial x}}{\\sqrt {1+\\left({\\frac {\\partial u}{\\partial x}}\\right)^{2}+\\left({\\frac {\\partial u}{\\partial y}}\\right)^{2}}}}+{\\frac {\\partial }{\\partial y}}{\\frac {\\frac {\\partial u}{\\partial y}}{\\sqrt {1+\\left({\\frac {\\partial u}{\\partial x}}\\right)^{2}+\\left({\\frac {\\partial u}{\\partial y}}\\right)^{2}}}}=0,}
then there are numbers a, b, and c with *u*(*x*, *y*) = *ax* + *by* + *c*.
In contrast to the earlier examples, this PDE is **nonlinear**, owing to the square roots and the squares. A **linear** PDE is one such that, if it is homogeneous, the sum of any two solutions is also a solution, and any constant multiple of any solution is also a solution.
## Definition
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A partial differential equation is an equation that involves an unknown function of n ≥ 2 {\\displaystyle n\\geq 2}  variables and (some of) its partial derivatives.[\[4\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-FOOTNOTEEvans19981%E2%80%932-4) That is, for the unknown function u : U → R , {\\displaystyle u:U\\rightarrow \\mathbb {R} ,}  of variables x \= ( x 1 , … , x n ) {\\displaystyle x=(x\_{1},\\dots ,x\_{n})}  belonging to the open subset U {\\displaystyle U}  of R n {\\displaystyle \\mathbb {R} ^{n}} , the k t h {\\displaystyle k^{th}} \-order partial differential equation is defined as F \[ D k u , D k − 1 u , … , D u , u , x \] \= 0 , {\\displaystyle F\[D^{k}u,D^{k-1}u,\\dots ,Du,u,x\]=0,} ![{\\displaystyle F\[D^{k}u,D^{k-1}u,\\dots ,Du,u,x\]=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6851c09712a689ae776fbfb75da28e8f95f39557) where F : R n k × R n k − 1 ⋯ × R n × R × U → R , {\\displaystyle F:\\mathbb {R} ^{n^{k}}\\times \\mathbb {R} ^{n^{k-1}}\\dots \\times \\mathbb {R} ^{n}\\times \\mathbb {R} \\times U\\rightarrow \\mathbb {R} ,}  and D {\\displaystyle D}  is the partial derivative operator.
### Notation
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Main article: [Notation for differentiation § Partial derivatives](https://en.wikipedia.org/wiki/Notation_for_differentiation#Partial_derivatives "Notation for differentiation")
When writing PDEs, it is common to denote partial derivatives using subscripts. For example: u x \= ∂ u ∂ x , u x x \= ∂ 2 u ∂ x 2 , u x y \= ∂ 2 u ∂ y ∂ x \= ∂ ∂ y ( ∂ u ∂ x ) . {\\displaystyle u\_{x}={\\frac {\\partial u}{\\partial x}},\\quad u\_{xx}={\\frac {\\partial ^{2}u}{\\partial x^{2}}},\\quad u\_{xy}={\\frac {\\partial ^{2}u}{\\partial y\\,\\partial x}}={\\frac {\\partial }{\\partial y}}\\left({\\frac {\\partial u}{\\partial x}}\\right).}  In the general situation that u is a function of n variables, then *u**i* denotes the first partial derivative relative to the i\-th input, *u**ij* denotes the second partial derivative relative to the i\-th and j\-th inputs, and so on.
The Greek letter Δ denotes the [Laplace operator](https://en.wikipedia.org/wiki/Laplace_operator "Laplace operator"); if u is a function of n variables, then Δ u \= u 11 \+ u 22 \+ ⋯ \+ u n n . {\\displaystyle \\Delta u=u\_{11}+u\_{22}+\\cdots +u\_{nn}.}  In the physics literature, the Laplace operator is often denoted by ∇2; in the mathematics literature, ∇2*u* may also denote the [Hessian matrix](https://en.wikipedia.org/wiki/Hessian_matrix "Hessian matrix") of u.
## Classification
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### Linear and nonlinear equations
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A PDE is called **linear** if it is linear in the unknown and its derivatives. For example, for a function u of x and y, a second order linear PDE is of the form a 1 ( x , y ) u x x \+ a 2 ( x , y ) u x y \+ a 3 ( x , y ) u y x \+ a 4 ( x , y ) u y y \+ a 5 ( x , y ) u x \+ a 6 ( x , y ) u y \+ a 7 ( x , y ) u \= f ( x , y ) {\\displaystyle a\_{1}(x,y)u\_{xx}+a\_{2}(x,y)u\_{xy}+a\_{3}(x,y)u\_{yx}+a\_{4}(x,y)u\_{yy}+a\_{5}(x,y)u\_{x}+a\_{6}(x,y)u\_{y}+a\_{7}(x,y)u=f(x,y)}  where *ai* and *f* are functions of the independent variables x and y only. (Often the mixed-partial derivatives *uxy* and *uyx* will be equated, but this is not required for the discussion of linearity.) If the *ai* are constants (independent of x and y) then the PDE is called **linear with constant coefficients**. If *f* is zero everywhere then the linear PDE is **homogeneous**, otherwise it is **inhomogeneous**. (This is separate from [asymptotic homogenization](https://en.wikipedia.org/wiki/Asymptotic_homogenization "Asymptotic homogenization"), which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.)
Nearest to linear PDEs are **semi-linear** PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order derivatives and the unknown function may appear arbitrarily. For example, a general second order semi-linear PDE in two variables is a 1 ( x , y ) u x x \+ a 2 ( x , y ) u x y \+ a 3 ( x , y ) u y x \+ a 4 ( x , y ) u y y \+ f ( u x , u y , u , x , y ) \= 0 {\\displaystyle a\_{1}(x,y)u\_{xx}+a\_{2}(x,y)u\_{xy}+a\_{3}(x,y)u\_{yx}+a\_{4}(x,y)u\_{yy}+f(u\_{x},u\_{y},u,x,y)=0} 
In a **quasilinear** PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives: a 1 ( u x , u y , u , x , y ) u x x \+ a 2 ( u x , u y , u , x , y ) u x y \+ a 3 ( u x , u y , u , x , y ) u y x \+ a 4 ( u x , u y , u , x , y ) u y y \+ f ( u x , u y , u , x , y ) \= 0 {\\displaystyle a\_{1}(u\_{x},u\_{y},u,x,y)u\_{xx}+a\_{2}(u\_{x},u\_{y},u,x,y)u\_{xy}+a\_{3}(u\_{x},u\_{y},u,x,y)u\_{yx}+a\_{4}(u\_{x},u\_{y},u,x,y)u\_{yy}+f(u\_{x},u\_{y},u,x,y)=0}  Many of the fundamental PDEs in physics are quasilinear, such as the [Einstein equations](https://en.wikipedia.org/wiki/Einstein_equations "Einstein equations") of [general relativity](https://en.wikipedia.org/wiki/General_relativity "General relativity") and the [Navier–Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations "Navier–Stokes equations") describing fluid motion.
A PDE without any linearity properties is called **fully [nonlinear](https://en.wikipedia.org/wiki/Nonlinear_partial_differential_equation "Nonlinear partial differential equation")**, and possesses nonlinearities on one or more of the highest-order derivatives. An example is the [Monge–Ampère equation](https://en.wikipedia.org/wiki/Monge%E2%80%93Amp%C3%A8re_equation "Monge–Ampère equation"), which arises in [differential geometry](https://en.wikipedia.org/wiki/Differential_geometry "Differential geometry").[\[5\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-PrincetonCompanion-5)
### Second order equations
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The elliptic/parabolic/hyperbolic classification provides a guide to appropriate [initial-](https://en.wikipedia.org/wiki/Initial_condition "Initial condition") and [boundary conditions](https://en.wikipedia.org/wiki/Boundary_value_problem "Boundary value problem") and to the [smoothness](https://en.wikipedia.org/wiki/Smoothness "Smoothness") of the solutions. Assuming *uxy* = *uyx*, the general linear second-order PDE in two independent variables has the form A u x x \+ 2 B u x y \+ C u y y \+ ⋯ (lower order terms) \= 0 , {\\displaystyle Au\_{xx}+2Bu\_{xy}+Cu\_{yy}+\\cdots {\\mbox{(lower order terms)}}=0,}  where the coefficients A, B, C... may depend upon x and y. If *A*2 + *B*2 + *C*2 \> 0 over a region of the xy\-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section: A x 2 \+ 2 B x y \+ C y 2 \+ ⋯ \= 0\. {\\displaystyle Ax^{2}+2Bxy+Cy^{2}+\\cdots =0.} 
More precisely, replacing ∂*x* by X, and likewise for other variables (formally this is done by a [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform")), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a [homogeneous polynomial](https://en.wikipedia.org/wiki/Homogeneous_polynomial "Homogeneous polynomial"), here a [quadratic form](https://en.wikipedia.org/wiki/Quadratic_form "Quadratic form")) being most significant for the classification.
Just as one classifies [conic sections](https://en.wikipedia.org/wiki/Conic_section "Conic section") and quadratic forms into parabolic, hyperbolic, and elliptic based on the [discriminant](https://en.wikipedia.org/wiki/Discriminant "Discriminant") *B*2 − 4*AC*, the same can be done for a second-order PDE at a given point. However, the [discriminant](https://en.wikipedia.org/wiki/Discriminant "Discriminant") in a PDE is given by *B*2 − *AC* due to the convention of the xy term being 2*B* rather than B; formally, the discriminant (of the associated quadratic form) is (2*B*)2 − 4*AC* = 4(*B*2 − *AC*), with the factor of 4 dropped for simplicity.
1. *B*2 − *AC* \< 0 (*[elliptic partial differential equation](https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation "Elliptic partial differential equation")*): Solutions of [elliptic PDEs](https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation "Elliptic partial differential equation") are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation") are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where *x* \< 0. By a change of variables, the equation can always be expressed in the form:
u
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u
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\=
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{\\displaystyle u\_{xx}+u\_{yy}+\\cdots =0,}
where x and y correspond to changed variables. This justifies [Laplace equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation") as an example of this type.[\[6\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-:0-6)
2. *B*2 − *AC* = 0 (*[parabolic partial differential equation](https://en.wikipedia.org/wiki/Parabolic_partial_differential_equation "Parabolic partial differential equation")*): Equations that are [parabolic](https://en.wikipedia.org/wiki/Parabolic_partial_differential_equation "Parabolic partial differential equation") at every point can be transformed into a form analogous to the [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation") by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where *x* = 0. By change of variables, the equation can always be expressed in the form:
u
x
x
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\=
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{\\displaystyle u\_{xx}+\\cdots =0,}
where x correspond to changed variables. This justifies the [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation"), which is of the form
u
t
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{\\textstyle u\_{t}-u\_{xx}+\\cdots =0}
, as an example of this type.[\[6\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-:0-6)
3. *B*2 − *AC* \> 0 (*[hyperbolic partial differential equation](https://en.wikipedia.org/wiki/Hyperbolic_partial_differential_equation "Hyperbolic partial differential equation")*): [hyperbolic](https://en.wikipedia.org/wiki/Hyperbolic_partial_differential_equation "Hyperbolic partial differential equation") equations retain any discontinuities of functions or derivatives in the initial data. An example is the [wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation"). The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where *x* \> 0. By change of variables, the equation can always be expressed in the form:
u
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{\\displaystyle u\_{xx}-u\_{yy}+\\cdots =0,}
where x and y correspond to changed variables. This justifies the [wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation") as an example of this type.[\[6\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-:0-6)
If there are n independent variables *x*1, *x*2 , …, *x**n*, a general linear partial differential equation of second order has the form L u \= ∑ i \= 1 n ∑ j \= 1 n a i , j ∂ 2 u ∂ x i ∂ x j \+ lower-order terms \= 0\. {\\displaystyle Lu=\\sum \_{i=1}^{n}\\sum \_{j=1}^{n}a\_{i,j}{\\frac {\\partial ^{2}u}{\\partial x\_{i}\\partial x\_{j}}}\\quad +{\\text{lower-order terms}}=0.} 
The classification depends upon the signature of the [eigenvalues](https://en.wikipedia.org/wiki/Eigenvalues "Eigenvalues") of the coefficient matrix *a**i*,*j*.
1. Elliptic: the eigenvalues are all positive or all negative.
2. Parabolic: the eigenvalues are all positive or all negative, except one that is zero.
3. Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
4. Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues.[\[7\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-7)
The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the [Laplace equation](https://en.wikipedia.org/wiki/Laplace_equation "Laplace equation"), the [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation"), and the [wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation").
However, the classification only depends on linearity of the second-order terms and is therefore applicable to semi- and quasilinear PDEs as well. The basic types also extend to hybrids such as the [Euler–Tricomi equation](https://en.wikipedia.org/wiki/Euler%E2%80%93Tricomi_equation "Euler–Tricomi equation"); varying from elliptic to hyperbolic for different [regions](https://en.wikipedia.org/wiki/Region_\(mathematics\) "Region (mathematics)") of the domain, as well as higher-order PDEs, but such knowledge is more specialized.
### Systems of first-order equations and characteristic surfaces
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See also: [First-order partial differential equation](https://en.wikipedia.org/wiki/First-order_partial_differential_equation "First-order partial differential equation")
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a [vector](https://en.wikipedia.org/wiki/Euclidean_vector "Euclidean vector") with m components, and the coefficient matrices Aν are m by m matrices for *ν* = 1, 2, …, *n*. The partial differential equation takes the form L u \= ∑ ν \= 1 n A ν ∂ u ∂ x ν \+ B \= 0 , {\\displaystyle Lu=\\sum \_{\\nu =1}^{n}A\_{\\nu }{\\frac {\\partial u}{\\partial x\_{\\nu }}}+B=0,}  where the coefficient matrices Aν and the vector B may depend upon x and u. If a [hypersurface](https://en.wikipedia.org/wiki/Hypersurface "Hypersurface") S is given in the implicit form φ ( x 1 , x 2 , … , x n ) \= 0 , {\\displaystyle \\varphi (x\_{1},x\_{2},\\ldots ,x\_{n})=0,}  where φ has a non-zero gradient, then S is a **characteristic surface** for the [operator](https://en.wikipedia.org/wiki/Differential_operator "Differential operator") L at a given point if the characteristic form vanishes: Q ( ∂ φ ∂ x 1 , … , ∂ φ ∂ x n ) \= det \[ ∑ ν \= 1 n A ν ∂ φ ∂ x ν \] \= 0\. {\\displaystyle Q\\left({\\frac {\\partial \\varphi }{\\partial x\_{1}}},\\ldots ,{\\frac {\\partial \\varphi }{\\partial x\_{n}}}\\right)=\\det \\left\[\\sum \_{\\nu =1}^{n}A\_{\\nu }{\\frac {\\partial \\varphi }{\\partial x\_{\\nu }}}\\right\]=0.} ![{\\displaystyle Q\\left({\\frac {\\partial \\varphi }{\\partial x\_{1}}},\\ldots ,{\\frac {\\partial \\varphi }{\\partial x\_{n}}}\\right)=\\det \\left\[\\sum \_{\\nu =1}^{n}A\_{\\nu }{\\frac {\\partial \\varphi }{\\partial x\_{\\nu }}}\\right\]=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fae8017409f092e75a9b50bf8e2febcdcdd407c)
The geometric interpretation of this condition is as follows: if data for u are prescribed on the surface S, then it may be possible to determine the normal derivative of u on S from the differential equation. If the data on S and the differential equation determine the normal derivative of u on S, then S is non-characteristic. If the data on S and the differential equation *do not* determine the normal derivative of u on S, then the surface is **characteristic**, and the differential equation restricts the data on S: the differential equation is *internal* to S.
1. A first-order system *Lu* = 0 is *elliptic* if no surface is characteristic for L: the values of u on S and the differential equation always determine the normal derivative of u on S.
2. A first-order system is *hyperbolic* at a point if there is a **spacelike** surface S with normal ξ at that point. This means that, given any non-trivial vector η orthogonal to ξ, and a scalar multiplier λ, the equation *Q*(*λξ* + *η*) = 0 has m real roots *λ*1, *λ*2, …, *λ**m*. The system is **strictly hyperbolic** if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form *Q*(*ζ*) = 0 defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has nm sheets, and the axis *ζ* = *λξ* runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.
## Analytical solutions
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### Separation of variables
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Main article: [Separable partial differential equation](https://en.wikipedia.org/wiki/Separable_partial_differential_equation "Separable partial differential equation")
Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a feature of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is *the* solution (this also applies to ODEs). We assume as an [ansatz](https://en.wikipedia.org/wiki/Ansatz "Ansatz") that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.[\[8\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-8)
In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.
This is possible for simple PDEs, which are called [separable partial differential equations](https://en.wikipedia.org/wiki/Separable_partial_differential_equation "Separable partial differential equation"), and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to [diagonal matrices](https://en.wikipedia.org/wiki/Diagonal_matrices "Diagonal matrices") – thinking of "the value for fixed x" as a coordinate, each coordinate can be understood separately.
This generalizes to the [method of characteristics](https://en.wikipedia.org/wiki/Method_of_characteristics "Method of characteristics"), and is also used in [integral transforms](https://en.wikipedia.org/wiki/Integral_transform "Integral transform").
### Method of characteristics
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Main article: [Method of characteristics](https://en.wikipedia.org/wiki/Method_of_characteristics "Method of characteristics")
The characteristic surface in *n* = *2*\-dimensional space is called a **characteristic curve**.[\[9\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-FOOTNOTEZachmanoglouThoe1986115%E2%80%93116-9) In special cases, one can find characteristic curves on which the first-order PDE reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the [method of characteristics](https://en.wikipedia.org/wiki/Method_of_characteristics "Method of characteristics").
More generally, applying the method to first-order PDEs in higher dimensions, one may find characteristic surfaces.
### Integral transform
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=11 "Edit section: Integral transform")\]
An [integral transform](https://en.wikipedia.org/wiki/Integral_transform "Integral transform") may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.
An important example of this is [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis"), which diagonalizes the heat equation using the [eigenbasis](https://en.wikipedia.org/wiki/Eigenbasis "Eigenbasis") of sinusoidal waves.
If the domain is finite or periodic, an infinite sum of solutions such as a [Fourier series](https://en.wikipedia.org/wiki/Fourier_series "Fourier series") is appropriate, but an integral of solutions such as a [Fourier integral](https://en.wikipedia.org/wiki/Fourier_integral "Fourier integral") is generally required for infinite domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral.
### Change of variables
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=12 "Edit section: Change of variables")\]
Often a PDE can be reduced to a simpler form with a known solution by a suitable [change of variables](https://en.wikipedia.org/wiki/Change_of_variables_\(PDE\) "Change of variables (PDE)"). For example, the [Black–Scholes equation](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation "Black–Scholes equation") ∂ V ∂ t \+ 1 2 σ 2 S 2 ∂ 2 V ∂ S 2 \+ r S ∂ V ∂ S − r V \= 0 {\\displaystyle {\\frac {\\partial V}{\\partial t}}+{\\tfrac {1}{2}}\\sigma ^{2}S^{2}{\\frac {\\partial ^{2}V}{\\partial S^{2}}}+rS{\\frac {\\partial V}{\\partial S}}-rV=0}  is reducible to the [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation") ∂ u ∂ τ \= ∂ 2 u ∂ x 2 {\\displaystyle {\\frac {\\partial u}{\\partial \\tau }}={\\frac {\\partial ^{2}u}{\\partial x^{2}}}}  by the change of variables[\[10\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-10) V ( S , t ) \= v ( x , τ ) , x \= ln ( S ) , τ \= 1 2 σ 2 ( T − t ) , v ( x , τ ) \= e − α x − β τ u ( x , τ ) . {\\displaystyle {\\begin{aligned}V(S,t)&=v(x,\\tau ),\\\\\[5px\]x&=\\ln \\left(S\\right),\\\\\[5px\]\\tau &={\\tfrac {1}{2}}\\sigma ^{2}(T-t),\\\\\[5px\]v(x,\\tau )&=e^{-\\alpha x-\\beta \\tau }u(x,\\tau ).\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}V(S,t)&=v(x,\\tau ),\\\\\[5px\]x&=\\ln \\left(S\\right),\\\\\[5px\]\\tau &={\\tfrac {1}{2}}\\sigma ^{2}(T-t),\\\\\[5px\]v(x,\\tau )&=e^{-\\alpha x-\\beta \\tau }u(x,\\tau ).\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e36d50d536184b7e1291e53a595db0b79fad14d9)
### Fundamental solution
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=13 "Edit section: Fundamental solution")\]
Main article: [Fundamental solution](https://en.wikipedia.org/wiki/Fundamental_solution "Fundamental solution")
Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding the [fundamental solution](https://en.wikipedia.org/wiki/Fundamental_solution "Fundamental solution") (the solution for a point source P ( D ) u \= δ {\\displaystyle P(D)u=\\delta } ), then taking the [convolution](https://en.wikipedia.org/wiki/Convolution "Convolution") with the boundary conditions to get the solution.
This is analogous in [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing") to understanding a filter by its [impulse response](https://en.wikipedia.org/wiki/Impulse_response "Impulse response").
### Superposition principle
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=14 "Edit section: Superposition principle")\]
Further information: [Superposition principle](https://en.wikipedia.org/wiki/Superposition_principle "Superposition principle")
The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example sin *x* + sin *x* = 2 sin *x*. The same principle can be observed in PDEs where the solutions may be real or complex and additive. If *u*1 and *u*2 are solutions of linear PDE in some function space R, then *u* = *c*1*u*1 + *c*2*u*2 with any constants *c*1 and *c*2 are also a solution of that PDE in the same function space.
### Methods for non-linear equations
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=15 "Edit section: Methods for non-linear equations")\]
See also: [nonlinear partial differential equation](https://en.wikipedia.org/wiki/Nonlinear_partial_differential_equation "Nonlinear partial differential equation")
There are no generally applicable analytical methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the [Cauchy–Kowalevski theorem](https://en.wikipedia.org/wiki/Cauchy%E2%80%93Kowalevski_theorem "Cauchy–Kowalevski theorem")) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of [analysis](https://en.wikipedia.org/wiki/Mathematical_analysis "Mathematical analysis")).
Nevertheless, some techniques can be used for several types of equations. The [h\-principle](https://en.wikipedia.org/wiki/H-principle "H-principle") is the most powerful method to solve [underdetermined](https://en.wikipedia.org/wiki/Underdetermined_system "Underdetermined system") equations. The [Riquier–Janet theory](https://en.wikipedia.org/w/index.php?title=Riquier%E2%80%93Janet_theory&action=edit&redlink=1 "Riquier–Janet theory (page does not exist)") is an effective method for obtaining information about many analytic [overdetermined](https://en.wikipedia.org/wiki/Overdetermined_system "Overdetermined system") systems.
The [method of characteristics](https://en.wikipedia.org/wiki/Method_of_characteristics "Method of characteristics") can be used in some very special cases to solve nonlinear partial differential equations.[\[11\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-11)
In some cases, a PDE can be solved via [perturbation analysis](https://en.wikipedia.org/wiki/Perturbation_analysis "Perturbation analysis") in which the solution is considered to be a correction to an equation with a known solution. Alternatives are [numerical analysis](https://en.wikipedia.org/wiki/Numerical_analysis "Numerical analysis") techniques from simple [finite difference](https://en.wikipedia.org/wiki/Finite_difference "Finite difference") schemes to the more mature [multigrid](https://en.wikipedia.org/wiki/Multigrid "Multigrid") and [finite element methods](https://en.wikipedia.org/wiki/Finite_element_method "Finite element method"). Many interesting problems in science and engineering are solved in this way using [computers](https://en.wikipedia.org/wiki/Computer "Computer"), sometimes high performance [supercomputers](https://en.wikipedia.org/wiki/Supercomputer "Supercomputer").
### Lie group method
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=16 "Edit section: Lie group method")\]
From 1870 [Sophus Lie](https://en.wikipedia.org/wiki/Sophus_Lie "Sophus Lie")'s work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called [Lie groups](https://en.wikipedia.org/wiki/Lie_group "Lie group"), be referred, to a common source; and that ordinary differential equations which admit the same [infinitesimal transformations](https://en.wikipedia.org/wiki/Infinitesimal_transformation "Infinitesimal transformation") present comparable difficulties of integration. He also emphasized the subject of [transformations of contact](https://en.wikipedia.org/wiki/Contact_transformation "Contact transformation").
A general approach to solving PDEs uses the symmetry property of differential equations, the continuous [infinitesimal transformations](https://en.wikipedia.org/wiki/Infinitesimal_transformation "Infinitesimal transformation") of solutions to solutions ([Lie theory](https://en.wikipedia.org/wiki/Lie_theory "Lie theory")). Continuous [group theory](https://en.wikipedia.org/wiki/Group_theory "Group theory"), [Lie algebras](https://en.wikipedia.org/wiki/Lie_algebras "Lie algebras") and [differential geometry](https://en.wikipedia.org/wiki/Differential_geometry "Differential geometry") are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its [Lax pairs](https://en.wikipedia.org/wiki/Lax_pair "Lax pair"), recursion operators, [Bäcklund transform](https://en.wikipedia.org/wiki/B%C3%A4cklund_transform "Bäcklund transform") and finally finding exact analytic solutions to the PDE.
Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.
### Semi-analytical methods
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=17 "Edit section: Semi-analytical methods")\]
The [Adomian decomposition method](https://en.wikipedia.org/wiki/Adomian_decomposition_method "Adomian decomposition method"),[\[12\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-12) the [Lyapunov](https://en.wikipedia.org/wiki/Aleksandr_Lyapunov "Aleksandr Lyapunov") artificial small parameter method, and his [homotopy perturbation method](https://en.wikipedia.org/wiki/Homotopy_perturbation_method "Homotopy perturbation method") are all special cases of the more general [homotopy analysis method](https://en.wikipedia.org/wiki/Homotopy_analysis_method "Homotopy analysis method").[\[13\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-13) These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known [perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory "Perturbation theory"), thus giving these methods greater flexibility and solution generality.
## Numerical solutions
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=18 "Edit section: Numerical solutions")\]
The three most widely used [numerical methods to solve PDEs](https://en.wikipedia.org/wiki/Numerical_partial_differential_equations "Numerical partial differential equations") are the [finite element method](https://en.wikipedia.org/wiki/Finite_element_analysis "Finite element analysis") (FEM), [finite volume methods](https://en.wikipedia.org/wiki/Finite_volume_method "Finite volume method") (FVM) and [finite difference methods](https://en.wikipedia.org/wiki/Finite_difference_method "Finite difference method") (FDM), as well other kind of methods called [meshfree methods](https://en.wikipedia.org/wiki/Meshfree_methods "Meshfree methods"), which were made to solve problems where the aforementioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version [hp-FEM](https://en.wikipedia.org/wiki/Hp-FEM "Hp-FEM"). Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM), [extended finite element method](https://en.wikipedia.org/wiki/Extended_finite_element_method "Extended finite element method") (XFEM), [spectral finite element method](https://en.wikipedia.org/wiki/Spectral_element_method "Spectral element method") (SFEM), [meshfree finite element method](https://en.wikipedia.org/wiki/Meshfree_methods "Meshfree methods"), [discontinuous Galerkin finite element method](https://en.wikipedia.org/wiki/Discontinuous_Galerkin_method "Discontinuous Galerkin method") (DGFEM), [element-free Galerkin method](https://en.wikipedia.org/w/index.php?title=Element-free_Galerkin_method&action=edit&redlink=1 "Element-free Galerkin method (page does not exist)") (EFGM), [interpolating element-free Galerkin method](https://en.wikipedia.org/w/index.php?title=Interpolating_element-free_Galerkin_method&action=edit&redlink=1 "Interpolating element-free Galerkin method (page does not exist)") (IEFGM), etc.
### Finite element method
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=19 "Edit section: Finite element method")\]
Main article: [Finite element method](https://en.wikipedia.org/wiki/Finite_element_method "Finite element method")
The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for approximating solutions of partial differential equations (PDE) as well as of integral equations using a finite set of functions.[\[14\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-14)[\[15\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-15) The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.
### Finite difference method
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=20 "Edit section: Finite difference method")\]
Main article: [Finite difference method](https://en.wikipedia.org/wiki/Finite_difference_method "Finite difference method")
Finite-difference methods are numerical methods for approximating the solutions to differential equations using [finite difference](https://en.wikipedia.org/wiki/Finite_difference "Finite difference") equations to approximate derivatives.
### Finite volume method
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=21 "Edit section: Finite volume method")\]
Main article: [Finite volume method](https://en.wikipedia.org/wiki/Finite_volume_method "Finite volume method")
Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the [divergence theorem](https://en.wikipedia.org/wiki/Divergence_theorem "Divergence theorem"). These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design.
### Neural networks
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=22 "Edit section: Neural networks")\]
This section is an excerpt from [Deep learning § Partial differential equations](https://en.wikipedia.org/wiki/Deep_learning#Partial_differential_equations "Deep learning").\[[edit](https://en.wikipedia.org/w/index.php?title=Deep_learning&action=edit)\]
Physics informed neural networks have been used to solve partial differential equations in both forward and inverse problems in a data driven manner.[\[16\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-16) One example is the reconstructing fluid flow governed by the [Navier-Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations "Navier–Stokes equations"). Using physics informed neural networks does not require the often expensive mesh generation that conventional [CFD](https://en.wikipedia.org/wiki/Computational_fluid_dynamics "Computational fluid dynamics") methods rely on.[\[17\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-17)[\[18\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-18) It is evident that geometric and physical constraints have a synergistic effect on neural PDE surrogates, thereby enhancing their efficacy in predicting stable and super long rollouts.[\[19\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-19)
## Weak solutions
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=23 "Edit section: Weak solutions")\]
Main article: [Weak solution](https://en.wikipedia.org/wiki/Weak_solution "Weak solution")
Weak solutions are functions that satisfy the PDE, yet in other meanings than regular sense. The meaning for this term may differ with context, and one of the most commonly used definitions is based on the notion of [distributions](https://en.wikipedia.org/wiki/Schwartz_distribution "Schwartz distribution").
An example[\[20\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-FOOTNOTEEvans1998chpt._6._Second-Order_Elliptic_Equations-20) for the definition of a weak solution is as follows:
Consider the boundary-value problem given by: L u \= f in U , u \= 0 on ∂ U , {\\displaystyle {\\begin{aligned}Lu&=f\\quad {\\text{in }}U,\\\\u&=0\\quad {\\text{on }}\\partial U,\\end{aligned}}}  where L u \= − ∑ i , j ∂ j ( a i j ∂ i u ) \+ ∑ i b i ∂ i u \+ c u {\\displaystyle Lu=-\\sum \_{i,j}\\partial \_{j}(a^{ij}\\partial \_{i}u)+\\sum \_{i}b^{i}\\partial \_{i}u+cu}  denotes a second-order partial differential operator in **divergence form**.
We say a [u ∈ H 0 1 ( U ) {\\displaystyle u\\in H\_{0}^{1}(U)} ](https://en.wikipedia.org/wiki/Sobolev_space "Sobolev space") is a weak solution if ∫ U \[ ∑ i , j a i j ( ∂ i u ) ( ∂ j v ) \+ ∑ i b i ( ∂ i u ) v \+ c u v \] d x \= ∫ U f v d x {\\displaystyle \\int \_{U}{\\bigg \[}\\sum \_{i,j}a^{ij}(\\partial \_{i}u)(\\partial \_{j}v)+\\sum \_{i}b^{i}(\\partial \_{i}u)v+cuv{\\bigg \]}dx=\\int \_{U}fvdx} ![{\\displaystyle \\int \_{U}{\\bigg \[}\\sum \_{i,j}a^{ij}(\\partial \_{i}u)(\\partial \_{j}v)+\\sum \_{i}b^{i}(\\partial \_{i}u)v+cuv{\\bigg \]}dx=\\int \_{U}fvdx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57cca9cc8062c43f270274e8eeba2a5e9ac9dfc8) for every v ∈ H 0 1 ( U ) {\\displaystyle v\\in H\_{0}^{1}(U)} , which can be derived by a formal integral by parts.
An example for a weak solution is as follows: ϕ ( x ) \= 1 4 π 1 \| x \| {\\displaystyle \\phi (x)={\\frac {1}{4\\pi }}{\\frac {1}{\|x\|}}}  is a weak solution satisfying ∇ 2 ϕ \= δ in R 3 {\\displaystyle \\nabla ^{2}\\phi =\\delta {\\text{ in }}R^{3}}  in distributional sense, as formally, ∫ R 3 ∇ 2 ϕ ( x ) ψ ( x ) d x \= ∫ R 3 ϕ ( x ) ∇ 2 ψ ( x ) d x \= ψ ( 0 ) for ψ ∈ C c ∞ ( R 3 ) . {\\displaystyle \\int \_{R^{3}}\\nabla ^{2}\\phi (x)\\psi (x)dx=\\int \_{R^{3}}\\phi (x)\\nabla ^{2}\\psi (x)dx=\\psi (0){\\text{ for }}\\psi \\in C\_{c}^{\\infty }(R^{3}).} 
## Theoretical studies
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In [pure mathematics](https://en.wikipedia.org/wiki/Pure_mathematics "Pure mathematics"), the theoretical studies of PDEs focus on the criteria for a solution to exist and the properties of a solution while finding its formula is often secondary.
### Well-posedness
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=25 "Edit section: Well-posedness")\]
Main article: [Well-posed problem](https://en.wikipedia.org/wiki/Well-posed_problem "Well-posed problem")
Well-posedness refers to a common schematic package of information about a PDE. To say that a PDE is well-posed, one must have:
- an existence and uniqueness theorem, asserting that by the prescription of some freely chosen functions, one can single out one specific solution of the PDE
- by [continuously](https://en.wikipedia.org/wiki/Continuity_\(mathematics\) "Continuity (mathematics)") changing the free choices, one continuously changes the corresponding solution
This is, by the necessity of being applicable to several different PDE, somewhat vague. The requirement of "continuity", in particular, is ambiguous, since there are usually many inequivalent means by which it can be rigorously defined. It is, however, somewhat unusual to study a PDE without specifying a way in which it is well-posed.
### Regularity
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=26 "Edit section: Regularity")\]
Main article: [Regularity theory](https://en.wikipedia.org/wiki/Regularity_theory "Regularity theory")
Regularity refers to the integrability and differentiability of weak solutions, which can often be represented by [Sobolev spaces](https://en.wikipedia.org/wiki/Sobolev_space "Sobolev space").
This problem arise due to the difficulty in searching for classical solutions. Researchers often tend to find weak solutions at first and then find out whether it is smooth enough to be qualified as a classical solution.
Results from [functional analysis](https://en.wikipedia.org/wiki/Functional_analysis "Functional analysis") are often used in this field of study.
## See also
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=27 "Edit section: See also")\]
**Some common PDEs**
- [Acoustic wave equation](https://en.wikipedia.org/wiki/Acoustic_wave_equation "Acoustic wave equation")
- [Burgers' equation](https://en.wikipedia.org/wiki/Burgers%27_equation "Burgers' equation")
- [Continuity equation](https://en.wikipedia.org/wiki/Continuity_equation "Continuity equation")
- [Heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation")
- [Helmholtz equation](https://en.wikipedia.org/wiki/Helmholtz_equation "Helmholtz equation")
- [Klein–Gordon equation](https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation "Klein–Gordon equation")
- [Jacobi equation](https://en.wikipedia.org/wiki/Jacobi_equation "Jacobi equation")
- [Lagrange equation](https://en.wikipedia.org/wiki/Lagrange_equation "Lagrange equation")
- [Lorenz equation](https://en.wikipedia.org/wiki/Lorenz_equation "Lorenz equation")
- [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation")
- [Maxwell's equations](https://en.wikipedia.org/wiki/Maxwell%27s_equations "Maxwell's equations")
- [Navier-Stokes equation](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations "Navier–Stokes equations")
- [Poisson's equation](https://en.wikipedia.org/wiki/Poisson%27s_equation "Poisson's equation")
- [Reaction–diffusion system](https://en.wikipedia.org/wiki/Reaction%E2%80%93diffusion_system "Reaction–diffusion system")
- [Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation")
- [Wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation")
**Types of boundary conditions**
- [Dirichlet boundary condition](https://en.wikipedia.org/wiki/Dirichlet_boundary_condition "Dirichlet boundary condition")
- [Neumann boundary condition](https://en.wikipedia.org/wiki/Neumann_boundary_condition "Neumann boundary condition")
- [Robin boundary condition](https://en.wikipedia.org/wiki/Robin_boundary_condition "Robin boundary condition")
- [Cauchy problem](https://en.wikipedia.org/wiki/Cauchy_problem "Cauchy problem")
**Various topics**
- [Jet bundle](https://en.wikipedia.org/wiki/Jet_bundle "Jet bundle")
- [Laplace transform applied to differential equations](https://en.wikipedia.org/wiki/Laplace_transform_applied_to_differential_equations "Laplace transform applied to differential equations")
- [List of dynamical systems and differential equations topics](https://en.wikipedia.org/wiki/List_of_dynamical_systems_and_differential_equations_topics "List of dynamical systems and differential equations topics")
- [Matrix differential equation](https://en.wikipedia.org/wiki/Matrix_differential_equation "Matrix differential equation")
- [Numerical partial differential equations](https://en.wikipedia.org/wiki/Numerical_partial_differential_equations "Numerical partial differential equations")
- [Partial differential algebraic equation](https://en.wikipedia.org/wiki/Partial_differential_algebraic_equation "Partial differential algebraic equation")
- [Recurrence relation](https://en.wikipedia.org/wiki/Recurrence_relation "Recurrence relation")
- [Stochastic processes and boundary value problems](https://en.wikipedia.org/wiki/Stochastic_processes_and_boundary_value_problems "Stochastic processes and boundary value problems")
## Notes
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=28 "Edit section: Notes")\]
1. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-1)**
["Regularity and singularities in elliptic PDE's: beyond monotonicity formulas \| EllipticPDE Project \| Fact Sheet \| H2020"](https://cordis.europa.eu/project/id/801867). *CORDIS \| European Commission*. Retrieved 2024-02-05.
2. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-2)**
[Klainerman, Sergiu](https://en.wikipedia.org/wiki/Sergiu_Klainerman "Sergiu Klainerman") (2010). "PDE as a Unified Subject". In Alon, N.; Bourgain, J.; Connes, A.; Gromov, M.; Milman, V. (eds.). *Visions in Mathematics*. Modern Birkhäuser Classics. Basel: Birkhäuser. pp. 279–315\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-0346-0422-2\_10](https://doi.org/10.1007%2F978-3-0346-0422-2_10). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-0346-0421-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-0346-0421-5 "Special:BookSources/978-3-0346-0421-5")
.
3. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-3)**
Erdoğan, M. Burak; Tzirakis, Nikolaos (2016). [*Dispersive Partial Differential Equations: Wellposedness and Applications*](https://www.cambridge.org/core/books/dispersive-partial-differential-equations/2DC65286BA080B54EB659E42A553CA88). London Mathematical Society Student Texts. Cambridge: Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-107-14904-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-14904-5 "Special:BookSources/978-1-107-14904-5")
.
4. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-FOOTNOTEEvans19981%E2%80%932_4-0)** [Evans 1998](https://en.wikipedia.org/wiki/Partial_differential_equation#CITEREFEvans1998), pp. 1–2.
5. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-PrincetonCompanion_5-0)**
Klainerman, Sergiu (2008), "Partial Differential Equations", in Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.), *The Princeton Companion to Mathematics*, Princeton University Press, pp. 455–483
6. ^ [***a***](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-:0_6-0) [***b***](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-:0_6-1) [***c***](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-:0_6-2)
Levandosky, Julie. ["Classification of Second-Order Equations"](https://web.stanford.edu/class/math220a/handouts/secondorder.pdf) (PDF).
7. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-7)** Courant and Hilbert (1962), p.182.
8. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-8)**
Gershenfeld, Neil (2000). [*The nature of mathematical modeling*](https://archive.org/details/naturemathematic00gers_334) (Reprinted (with corr.) ed.). Cambridge: Cambridge University Press. p. [27](https://archive.org/details/naturemathematic00gers_334/page/n32). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0521570956](https://en.wikipedia.org/wiki/Special:BookSources/0521570956 "Special:BookSources/0521570956")
.
9. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-FOOTNOTEZachmanoglouThoe1986115%E2%80%93116_9-0)** [Zachmanoglou & Thoe 1986](https://en.wikipedia.org/wiki/Partial_differential_equation#CITEREFZachmanoglouThoe1986), pp. 115–116.
10. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-10)**
Wilmott, Paul; Howison, Sam; Dewynne, Jeff (1995). [*The Mathematics of Financial Derivatives*](https://books.google.com/books?id=VYVhnC3fIVEC&pg=PA76). Cambridge University Press. pp. 76–81\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-521-49789-2](https://en.wikipedia.org/wiki/Special:BookSources/0-521-49789-2 "Special:BookSources/0-521-49789-2")
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11. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-11)**
Logan, J. David (1994). "First Order Equations and Characteristics". *An Introduction to Nonlinear Partial Differential Equations*. New York: John Wiley & Sons. pp. 51–79\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-471-59916-6](https://en.wikipedia.org/wiki/Special:BookSources/0-471-59916-6 "Special:BookSources/0-471-59916-6")
.
12. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-12)**
[Adomian, G.](https://en.wikipedia.org/wiki/George_Adomian "George Adomian") (1994). [*Solving Frontier problems of Physics: The decomposition method*](https://books.google.com/books?id=UKPqCAAAQBAJ&q=%22partial+differential%22). Kluwer Academic Publishers. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9789401582896](https://en.wikipedia.org/wiki/Special:BookSources/9789401582896 "Special:BookSources/9789401582896")
.
13. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-13)**
[Liao, S. J.](https://en.wikipedia.org/wiki/Liao_Shijun "Liao Shijun") (2003). *Beyond Perturbation: Introduction to the Homotopy Analysis Method*. Boca Raton: Chapman & Hall/ CRC Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[1-58488-407-X](https://en.wikipedia.org/wiki/Special:BookSources/1-58488-407-X "Special:BookSources/1-58488-407-X")
.
14. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-14)**
Solin, P. (2005). *Partial Differential Equations and the Finite Element Method*. Hoboken, New Jersey: J. Wiley & Sons. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-471-72070-4](https://en.wikipedia.org/wiki/Special:BookSources/0-471-72070-4 "Special:BookSources/0-471-72070-4")
.
15. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-15)**
Solin, P.; Segeth, K. & Dolezel, I. (2003). *Higher-Order Finite Element Methods*. Boca Raton: Chapman & Hall/CRC Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[1-58488-438-X](https://en.wikipedia.org/wiki/Special:BookSources/1-58488-438-X "Special:BookSources/1-58488-438-X")
.
16. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-16)**
Raissi, M.; Perdikaris, P.; Karniadakis, G. E. (2019-02-01). ["Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations"](https://doi.org/10.1016%2Fj.jcp.2018.10.045). *Journal of Computational Physics*. **378**: 686–707\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2019JCoPh.378..686R](https://ui.adsabs.harvard.edu/abs/2019JCoPh.378..686R). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.jcp.2018.10.045](https://doi.org/10.1016%2Fj.jcp.2018.10.045). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0021-9991](https://search.worldcat.org/issn/0021-9991). [OSTI](https://en.wikipedia.org/wiki/OSTI_\(identifier\) "OSTI (identifier)") [1595805](https://www.osti.gov/biblio/1595805). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [57379996](https://api.semanticscholar.org/CorpusID:57379996).
17. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-17)**
Mao, Zhiping; Jagtap, Ameya D.; Karniadakis, George Em (2020-03-01). ["Physics-informed neural networks for high-speed flows"](https://doi.org/10.1016%2Fj.cma.2019.112789). *Computer Methods in Applied Mechanics and Engineering*. **360** 112789. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2020CMAME.360k2789M](https://ui.adsabs.harvard.edu/abs/2020CMAME.360k2789M). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.cma.2019.112789](https://doi.org/10.1016%2Fj.cma.2019.112789). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0045-7825](https://search.worldcat.org/issn/0045-7825). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [212755458](https://api.semanticscholar.org/CorpusID:212755458).
18. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-18)**
Raissi, Maziar; Yazdani, Alireza; Karniadakis, George Em (2020-02-28). ["Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7219083). *Science*. **367** (6481): 1026–1030\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2020Sci...367.1026R](https://ui.adsabs.harvard.edu/abs/2020Sci...367.1026R). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1126/science.aaw4741](https://doi.org/10.1126%2Fscience.aaw4741). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [7219083](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7219083). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [32001523](https://pubmed.ncbi.nlm.nih.gov/32001523).
19. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-19)** Huang, Yunfei and Greenberg, David S. "[Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogates](https://arxiv.org/pdf/2506.05513)." Proceedings of the 42th international conference on Machine learning. ACM, 2025.
20. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-FOOTNOTEEvans1998chpt._6._Second-Order_Elliptic_Equations_20-0)** [Evans 1998](https://en.wikipedia.org/wiki/Partial_differential_equation#CITEREFEvans1998), chpt. 6. Second-Order Elliptic Equations.
## References
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=29 "Edit section: References")\]
- [Courant, R.](https://en.wikipedia.org/wiki/Richard_Courant "Richard Courant") & [Hilbert, D.](https://en.wikipedia.org/wiki/David_Hilbert "David Hilbert") (1962), [*Methods of Mathematical Physics*](https://books.google.com/books?id=fcZV4ohrerwC), vol. II, New York: Wiley-Interscience, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9783527617241](https://en.wikipedia.org/wiki/Special:BookSources/9783527617241 "Special:BookSources/9783527617241")
`{{citation}}`: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date "Help:CS1 errors")).
- Drábek, Pavel; Holubová, Gabriela (2007). *Elements of partial differential equations* (Online ed.). Berlin: de Gruyter. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9783110191240](https://en.wikipedia.org/wiki/Special:BookSources/9783110191240 "Special:BookSources/9783110191240")
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- [Evans, Lawrence C.](https://en.wikipedia.org/wiki/Lawrence_C._Evans "Lawrence C. Evans") (1998). [*Partial differential equations*](https://math24.wordpress.com/wp-content/uploads/2013/02/partial-differential-equations-by-evans.pdf) (PDF). Providence (R. I.): American mathematical society. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-8218-0772-2](https://en.wikipedia.org/wiki/Special:BookSources/0-8218-0772-2 "Special:BookSources/0-8218-0772-2")
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- [Ibragimov, Nail H.](https://en.wikipedia.org/wiki/Nail_H._Ibragimov "Nail H. Ibragimov") (1993), *CRC Handbook of Lie Group Analysis of Differential Equations Vol. 1-3*, Providence: CRC-Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-8493-4488-3](https://en.wikipedia.org/wiki/Special:BookSources/0-8493-4488-3 "Special:BookSources/0-8493-4488-3")
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- [John, F.](https://en.wikipedia.org/wiki/Fritz_John "Fritz John") (1982), [*Partial Differential Equations*](https://archive.org/details/partialdifferent00john_0) (4th ed.), New York: Springer-Verlag, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-387-90609-6](https://en.wikipedia.org/wiki/Special:BookSources/0-387-90609-6 "Special:BookSources/0-387-90609-6")
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- Gallouët, T.; Herbin, R. (2025). [*Weak Solutions to Partial Differential Equations*](https://link.springer.com/book/10.1007/978-3-031-98982-7). New York: Springer-Verlag. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-031-98981-0](https://en.wikipedia.org/wiki/Special:BookSources/978-3-031-98981-0 "Special:BookSources/978-3-031-98981-0")
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- [Jost, J.](https://en.wikipedia.org/wiki/J%C3%BCrgen_Jost "Jürgen Jost") (2002), *Partial Differential Equations*, New York: Springer-Verlag, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-387-95428-7](https://en.wikipedia.org/wiki/Special:BookSources/0-387-95428-7 "Special:BookSources/0-387-95428-7")
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- [Olver, P.J.](https://en.wikipedia.org/wiki/Peter_J._Olver "Peter J. Olver") (1995), *Equivalence, Invariants and Symmetry*, Cambridge Press
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- [Petrovskii, I. G.](https://en.wikipedia.org/wiki/Ivan_Petrovsky "Ivan Petrovsky") (1967), *Partial Differential Equations*, Philadelphia: W. B. Saunders Co.
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- Pinchover, Y. & Rubinstein, J. (2005), *An Introduction to Partial Differential Equations*, New York: Cambridge University Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-521-84886-5](https://en.wikipedia.org/wiki/Special:BookSources/0-521-84886-5 "Special:BookSources/0-521-84886-5")
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- [Polyanin, A. D.](https://en.wikipedia.org/wiki/Andrei_Polyanin "Andrei Polyanin") (2002), *Handbook of Linear Partial Differential Equations for Engineers and Scientists*, Boca Raton: Chapman & Hall/CRC Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[1-58488-299-9](https://en.wikipedia.org/wiki/Special:BookSources/1-58488-299-9 "Special:BookSources/1-58488-299-9")
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- [Polyanin, A. D.](https://en.wikipedia.org/wiki/Andrei_Polyanin "Andrei Polyanin") & Zaitsev, V. F. (2004), *Handbook of Nonlinear Partial Differential Equations*, Boca Raton: Chapman & Hall/CRC Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[1-58488-355-3](https://en.wikipedia.org/wiki/Special:BookSources/1-58488-355-3 "Special:BookSources/1-58488-355-3")
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- [Polyanin, A. D.](https://en.wikipedia.org/wiki/Andrei_Polyanin "Andrei Polyanin"); Zaitsev, V. F. & Moussiaux, A. (2002), *Handbook of First Order Partial Differential Equations*, London: Taylor & Francis, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-415-27267-X](https://en.wikipedia.org/wiki/Special:BookSources/0-415-27267-X "Special:BookSources/0-415-27267-X")
.
- Roubíček, T. (2013), [*Nonlinear Partial Differential Equations with Applications*](https://cds.cern.ch/record/880983/files/9783764372934_TOC.pdf) (PDF), International Series of Numerical Mathematics, vol. 153 (2nd ed.), Basel, Boston, Berlin: Birkhäuser, [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-0348-0513-1](https://doi.org/10.1007%2F978-3-0348-0513-1), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-0348-0512-4](https://en.wikipedia.org/wiki/Special:BookSources/978-3-0348-0512-4 "Special:BookSources/978-3-0348-0512-4")
, [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [3014456](https://mathscinet.ams.org/mathscinet-getitem?mr=3014456)
- [Stephani, H.](https://en.wikipedia.org/wiki/Hans_Stephani "Hans Stephani") (1989), MacCallum, M. (ed.), *Differential Equations: Their Solution Using Symmetries*, Cambridge University Press
.
- Wazwaz, Abdul-Majid (2009). *Partial Differential Equations and Solitary Waves Theory*. Higher Education Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-642-00251-9](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-00251-9 "Special:BookSources/978-3-642-00251-9")
.
- Wazwaz, Abdul-Majid (2002). *Partial Differential Equations Methods and Applications*. A.A. Balkema. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[90-5809-369-7](https://en.wikipedia.org/wiki/Special:BookSources/90-5809-369-7 "Special:BookSources/90-5809-369-7")
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- Zwillinger, D. (1997), *Handbook of Differential Equations* (3rd ed.), Boston: Academic Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-12-784395-7](https://en.wikipedia.org/wiki/Special:BookSources/0-12-784395-7 "Special:BookSources/0-12-784395-7")
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- [Gershenfeld, N.](https://en.wikipedia.org/wiki/Neil_Gershenfeld "Neil Gershenfeld") (1999), *The Nature of Mathematical Modeling* (1st ed.), New York: Cambridge University Press, New York, NY, USA, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-521-57095-6](https://en.wikipedia.org/wiki/Special:BookSources/0-521-57095-6 "Special:BookSources/0-521-57095-6")
.
- Krasil'shchik, I.S. & [Vinogradov, A.M., Eds.](https://en.wikipedia.org/wiki/Alexandre_Mikhailovich_Vinogradov "Alexandre Mikhailovich Vinogradov") (1999), *Symmetries and Conservation Laws for Differential Equations of Mathematical Physics*, American Mathematical Society, Providence, Rhode Island, USA, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-8218-0958-X](https://en.wikipedia.org/wiki/Special:BookSources/0-8218-0958-X "Special:BookSources/0-8218-0958-X")
`{{citation}}`: CS1 maint: multiple names: authors list ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_multiple_names:_authors_list "Category:CS1 maint: multiple names: authors list")).
- Krasil'shchik, I.S.; Lychagin, V.V. & [Vinogradov, A.M.](https://en.wikipedia.org/wiki/Alexandre_Mikhailovich_Vinogradov "Alexandre Mikhailovich Vinogradov") (1986), *Geometry of Jet Spaces and Nonlinear Partial Differential Equations*, Gordon and Breach Science Publishers, New York, London, Paris, Montreux, Tokyo, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[2-88124-051-8](https://en.wikipedia.org/wiki/Special:BookSources/2-88124-051-8 "Special:BookSources/2-88124-051-8")
.
- [Vinogradov, A.M.](https://en.wikipedia.org/wiki/Alexandre_Mikhailovich_Vinogradov "Alexandre Mikhailovich Vinogradov") (2001), *Cohomological Analysis of Partial Differential Equations and Secondary Calculus*, American Mathematical Society, Providence, Rhode Island, USA, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-8218-2922-X](https://en.wikipedia.org/wiki/Special:BookSources/0-8218-2922-X "Special:BookSources/0-8218-2922-X")
.
- [Gustafsson, Bertil](https://en.wikipedia.org/wiki/Bertil_Gustafsson "Bertil Gustafsson") (2008). *High Order Difference Methods for Time Dependent PDE*. Springer Series in Computational Mathematics. Vol. 38. Springer. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-540-74993-6](https://doi.org/10.1007%2F978-3-540-74993-6). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-3-540-74992-9](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-74992-9 "Special:BookSources/978-3-540-74992-9")
.
- Zachmanoglou, E. C.; Thoe, Dale W. (1986). *Introduction to Partial Differential Equations with Applications*. New York: Courier Corporation. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-486-65251-3](https://en.wikipedia.org/wiki/Special:BookSources/0-486-65251-3 "Special:BookSources/0-486-65251-3")
.
## Further reading
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=30 "Edit section: Further reading")\]
- [Cajori, Florian](https://en.wikipedia.org/wiki/Florian_Cajori "Florian Cajori") (1928). ["The Early History of Partial Differential Equations and of Partial Differentiation and Integration"](https://web.archive.org/web/20181123102253/http://www.math.harvard.edu/archive/21a_fall_14/exhibits/cajori/cajori.pdf) (PDF). *The American Mathematical Monthly*. **35** (9): 459–467\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2307/2298771](https://doi.org/10.2307%2F2298771). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [2298771](https://www.jstor.org/stable/2298771). Archived from [the original](http://www.math.harvard.edu/archive/21a_fall_14/exhibits/cajori/cajori.pdf) (PDF) on 2018-11-23. Retrieved 2016-05-15.
- [Nirenberg, Louis](https://en.wikipedia.org/wiki/Louis_Nirenberg "Louis Nirenberg") (1994). "Partial differential equations in the first half of the century." Development of mathematics 1900–1950 (Luxembourg, 1992), 479–515, Birkhäuser, Basel.
- [Brezis, Haïm](https://en.wikipedia.org/wiki/Ha%C3%AFm_Brezis "Haïm Brezis"); [Browder, Felix](https://en.wikipedia.org/wiki/Felix_Browder "Felix Browder") (1998). ["Partial Differential Equations in the 20th Century"](https://doi.org/10.1006%2Faima.1997.1713). *[Advances in Mathematics](https://en.wikipedia.org/wiki/Advances_in_Mathematics "Advances in Mathematics")*. **135** (1): 76–144\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1006/aima.1997.1713](https://doi.org/10.1006%2Faima.1997.1713).
## External links
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=31 "Edit section: External links")\]
**Partial differential equation** at Wikipedia's [sister projects](https://en.wikipedia.org/wiki/Wikipedia:Wikimedia_sister_projects "Wikipedia:Wikimedia sister projects")
- [](https://en.wikipedia.org/wiki/File:Commons-logo.svg)[Media](https://commons.wikimedia.org/wiki/Category:Solutions_of_PDE "c:Category:Solutions of PDE") from Commons
- [Quotations](https://en.wikiquote.org/wiki/Partial_differential_equation "q:Partial differential equation") from Wikiquote
- [](https://en.wikipedia.org/wiki/File:Wikibooks-logo.svg)[Textbooks](https://en.wikibooks.org/wiki/Partial_Differential_Equations "b:Partial Differential Equations") from Wikibooks
- ["Differential equation, partial"](https://www.encyclopediaofmath.org/index.php?title=Differential_equation,_partial), *[Encyclopedia of Mathematics](https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics "Encyclopedia of Mathematics")*, [EMS Press](https://en.wikipedia.org/wiki/European_Mathematical_Society "European Mathematical Society"), 2001 \[1994\]
- [Partial Differential Equations: Exact Solutions](http://eqworld.ipmnet.ru/en/pde-en.htm) at EqWorld: The World of Mathematical Equations.
- [Partial Differential Equations: Index](http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-pde.htm) at EqWorld: The World of Mathematical Equations.
- [Partial Differential Equations: Methods](http://eqworld.ipmnet.ru/en/methods/meth-pde.htm) at EqWorld: The World of Mathematical Equations.
- [Example problems with solutions](https://web.archive.org/web/20170701144823/http://www.exampleproblems.com/wiki/index.php?title=Partial_Differential_Equations) at exampleproblems.com
- [Partial Differential Equations](http://mathworld.wolfram.com/PartialDifferentialEquation.html) at mathworld.wolfram.com
- [Partial Differential Equations](https://reference.wolfram.com/language/FEMDocumentation/tutorial/SolvingPDEwithFEM.html) with Mathematica
- [Partial Differential Equations](http://www.mathworks.com/moler/pdes.pdf) [Archived](https://web.archive.org/web/20160817082230/http://www.mathworks.com/moler/pdes.pdf) 2016-08-17 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine") in Cleve Moler: Numerical Computing with MATLAB
- [Partial Differential Equations](http://www.nag.com/numeric/fl/nagdoc_fl24/html/D03/d03intro.html) at nag.com
- Sanderson, Grant (April 21, 2019). ["But what is a partial differential equation?"](https://www.youtube.com/watch?v=ly4S0oi3Yz8&list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6). *3Blue1Brown*. [Archived](https://ghostarchive.org/varchive/youtube/20211102/ly4S0oi3Yz8) from the original on 2021-11-02 – via [YouTube](https://en.wikipedia.org/wiki/YouTube "YouTube").
- [Solutions faibles des équations différentielles](https://hal.science/cel-01196782) at hal.science
| [v](https://en.wikipedia.org/wiki/Template:Differential_equations_topics "Template:Differential equations topics") [t](https://en.wikipedia.org/wiki/Template_talk:Differential_equations_topics "Template talk:Differential equations topics") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Differential_equations_topics "Special:EditPage/Template:Differential equations topics")[Differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation") | |
|---|---|
| Classification | |
| | |
| Operations | [Differential operator](https://en.wikipedia.org/wiki/Differential_operator "Differential operator") [Notation for differentiation](https://en.wikipedia.org/wiki/Notation_for_differentiation "Notation for differentiation") [Ordinary](https://en.wikipedia.org/wiki/Ordinary_differential_equation "Ordinary differential equation") [Partial]() [Differential-algebraic](https://en.wikipedia.org/wiki/Differential-algebraic_system_of_equations "Differential-algebraic system of equations") [Integro-differential](https://en.wikipedia.org/wiki/Integro-differential_equation "Integro-differential equation") [Fractional](https://en.wikipedia.org/wiki/Fractional_differential_equations "Fractional differential equations") [Linear](https://en.wikipedia.org/wiki/Linear_differential_equation "Linear differential equation") [Non-linear](https://en.wikipedia.org/wiki/Nonlinear_system#Nonlinear_differential_equations "Nonlinear system") [Holonomic](https://en.wikipedia.org/wiki/Holonomic_function "Holonomic function") |
| Attributes of variables | [Dependent and independent variables](https://en.wikipedia.org/wiki/Dependent_and_independent_variables "Dependent and independent variables") [Homogeneous](https://en.wikipedia.org/wiki/Homogeneous_differential_equation "Homogeneous differential equation") [Nonhomogeneous](https://en.wikipedia.org/wiki/Non-homogeneous_differential_equation "Non-homogeneous differential equation") [Coupled](https://en.wikipedia.org/wiki/Differential_equation "Differential equation") [Decoupled](https://en.wikipedia.org/wiki/Differential_equation "Differential equation") [Order](https://en.wikipedia.org/wiki/Differential_equation "Differential equation") [Degree](https://en.wikipedia.org/wiki/Differential_equation "Differential equation") [Autonomous](https://en.wikipedia.org/wiki/Autonomous_system_\(mathematics\) "Autonomous system (mathematics)") [Exact differential equation](https://en.wikipedia.org/wiki/Exact_differential_equation "Exact differential equation") [On jet bundles](https://en.wikipedia.org/wiki/Jet_bundle#Partial_differential_equations "Jet bundle") |
| Relation to processes | [Difference](https://en.wikipedia.org/wiki/Difference_equation "Difference equation") (discrete analogue) [Stochastic](https://en.wikipedia.org/wiki/Stochastic_differential_equation "Stochastic differential equation") [Stochastic partial](https://en.wikipedia.org/wiki/Stochastic_partial_differential_equation "Stochastic partial differential equation") [Delay](https://en.wikipedia.org/wiki/Delay_differential_equation "Delay differential equation") |
| Solutions | |
| | |
| Existence/uniqueness | [Picard–Lindelöf theorem](https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem "Picard–Lindelöf theorem") [Peano existence theorem](https://en.wikipedia.org/wiki/Peano_existence_theorem "Peano existence theorem") [Carathéodory's existence theorem](https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_existence_theorem "Carathéodory's existence theorem") [Cauchy–Kowalevski theorem](https://en.wikipedia.org/wiki/Cauchy%E2%80%93Kowalevski_theorem "Cauchy–Kowalevski theorem") |
| Solution topics | [Wronskian](https://en.wikipedia.org/wiki/Wronskian "Wronskian") [Phase portrait](https://en.wikipedia.org/wiki/Phase_portrait "Phase portrait") [Phase space](https://en.wikipedia.org/wiki/Phase_space "Phase space") [Lyapunov stability](https://en.wikipedia.org/wiki/Lyapunov_stability "Lyapunov stability") [Asymptotic stability](https://en.wikipedia.org/wiki/Asymptotic_stability "Asymptotic stability") [Exponential stability](https://en.wikipedia.org/wiki/Exponential_stability "Exponential stability") [Rate of convergence](https://en.wikipedia.org/wiki/Rate_of_convergence "Rate of convergence") [Series solutions](https://en.wikipedia.org/wiki/Power_series_solution_of_differential_equations "Power series solution of differential equations") [Integral](https://en.wikipedia.org/wiki/Integral "Integral") solutions [Numerical integration](https://en.wikipedia.org/wiki/Numerical_integration "Numerical integration") [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function "Dirac delta function") |
| Solution methods | [Inspection](https://en.wikipedia.org/wiki/List_of_mathematical_jargon#Proof_techniques "List of mathematical jargon") [Substitution](https://en.wikipedia.org/wiki/Integration_by_substitution "Integration by substitution") [Separation of variables](https://en.wikipedia.org/wiki/Separation_of_variables "Separation of variables") [Method of undetermined coefficients](https://en.wikipedia.org/wiki/Method_of_undetermined_coefficients "Method of undetermined coefficients") [Variation of parameters](https://en.wikipedia.org/wiki/Variation_of_parameters "Variation of parameters") [Integrating factor](https://en.wikipedia.org/wiki/Integrating_factor "Integrating factor") [Integral transforms](https://en.wikipedia.org/wiki/Integral_transform "Integral transform") [Euler method](https://en.wikipedia.org/wiki/Euler_method "Euler method") [Finite difference method](https://en.wikipedia.org/wiki/Finite_difference_method "Finite difference method") [Crank–Nicolson method](https://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method "Crank–Nicolson method") [Runge–Kutta methods](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods "Runge–Kutta methods") [Finite element method](https://en.wikipedia.org/wiki/Finite_element_method "Finite element method") [Finite volume method](https://en.wikipedia.org/wiki/Finite_volume_method "Finite volume method") [Galerkin method](https://en.wikipedia.org/wiki/Galerkin_method "Galerkin method") [Perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory "Perturbation theory") |
| Examples | [List of named differential equations](https://en.wikipedia.org/wiki/List_of_named_differential_equations "List of named differential equations") [List of linear ordinary differential equations](https://en.wikipedia.org/wiki/List_of_linear_ordinary_differential_equations "List of linear ordinary differential equations") [List of nonlinear ordinary differential equations](https://en.wikipedia.org/wiki/List_of_nonlinear_ordinary_differential_equations "List of nonlinear ordinary differential equations") [List of nonlinear partial differential equations](https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations "List of nonlinear partial differential equations") |
| Mathematicians | [Isaac Newton](https://en.wikipedia.org/wiki/Isaac_Newton "Isaac Newton") [Gottfried Wilhelm Leibniz](https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz "Gottfried Wilhelm Leibniz") [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") [Jacob Bernoulli](https://en.wikipedia.org/wiki/Jacob_Bernoulli "Jacob Bernoulli") [Émile Picard](https://en.wikipedia.org/wiki/%C3%89mile_Picard "Émile Picard") [Józef Maria Hoene-Wroński](https://en.wikipedia.org/wiki/J%C3%B3zef_Maria_Hoene-Wro%C5%84ski "Józef Maria Hoene-Wroński") [Ernst Lindelöf](https://en.wikipedia.org/wiki/Ernst_Leonard_Lindel%C3%B6f "Ernst Leonard Lindelöf") [Rudolf Lipschitz](https://en.wikipedia.org/wiki/Rudolf_Lipschitz "Rudolf Lipschitz") [Joseph-Louis Lagrange](https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange "Joseph-Louis Lagrange") [Augustin-Louis Cauchy](https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy "Augustin-Louis Cauchy") [John Crank](https://en.wikipedia.org/wiki/John_Crank "John Crank") [Phyllis Nicolson](https://en.wikipedia.org/wiki/Phyllis_Nicolson "Phyllis Nicolson") [Carl David Tolmé Runge](https://en.wikipedia.org/wiki/Carl_David_Tolm%C3%A9_Runge "Carl David Tolmé Runge") [Martin Kutta](https://en.wikipedia.org/wiki/Martin_Kutta "Martin Kutta") [Sofya Kovalevskaya](https://en.wikipedia.org/wiki/Sofya_Kovalevskaya "Sofya Kovalevskaya") |
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Partial differential equation
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| Readable Markdown | [](https://en.wikipedia.org/wiki/File:Heat.gif)
A visualisation of a solution to the two-dimensional [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation") with temperature represented by the vertical direction and color
In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), a **partial differential equation** (**PDE**) is an equation which involves a [multivariable function](https://en.wikipedia.org/wiki/Function_of_several_real_variables "Function of several real variables") and one or more of its [partial derivatives](https://en.wikipedia.org/wiki/Partial_derivative "Partial derivative").
The function is often thought of as an "unknown" that solves the equation. However, it is often impossible to write down explicit formulas for solutions of partial differential equations. Hence there is a vast amount of modern mathematical and scientific research on methods to [numerically approximate](https://en.wikipedia.org/wiki/Numerical_methods_for_partial_differential_equations "Numerical methods for partial differential equations") solutions of partial differential equations using computers. Partial differential equations also occupy a large sector of [pure mathematical research](https://en.wikipedia.org/wiki/Pure_mathematics "Pure mathematics"), where the focus is on the qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability.[\[1\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-1) Among the many open questions are the [existence and smoothness](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness "Navier–Stokes existence and smoothness") of solutions to the [Navier–Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations "Navier–Stokes equations"), named as one of the [Millennium Prize Problems](https://en.wikipedia.org/wiki/Millennium_Prize_Problems "Millennium Prize Problems") in 2000.
Partial differential equations occur very widely in mathematically oriented scientific fields, such as [physics](https://en.wikipedia.org/wiki/Physics "Physics") and [engineering](https://en.wikipedia.org/wiki/Engineering "Engineering"). For instance, they are foundational in the modern scientific understanding of [sound](https://en.wikipedia.org/wiki/Sound "Sound"), [heat](https://en.wikipedia.org/wiki/Heat "Heat"), [diffusion](https://en.wikipedia.org/wiki/Diffusion "Diffusion"), [electrostatics](https://en.wikipedia.org/wiki/Electrostatics "Electrostatics"), [electrodynamics](https://en.wikipedia.org/wiki/Electromagnetism "Electromagnetism"), [thermodynamics](https://en.wikipedia.org/wiki/Thermodynamics "Thermodynamics"), [fluid dynamics](https://en.wikipedia.org/wiki/Fluid_dynamics "Fluid dynamics"), [elasticity](https://en.wikipedia.org/wiki/Elasticity_\(physics\) "Elasticity (physics)"), [general relativity](https://en.wikipedia.org/wiki/General_relativity "General relativity"), and [quantum mechanics](https://en.wikipedia.org/wiki/Quantum_mechanics "Quantum mechanics") ([Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation"), [Pauli equation](https://en.wikipedia.org/wiki/Pauli_equation "Pauli equation") etc.). They also arise from many purely mathematical considerations, such as [differential geometry](https://en.wikipedia.org/wiki/Differential_geometry "Differential geometry") and the [calculus of variations](https://en.wikipedia.org/wiki/Calculus_of_variations "Calculus of variations"); among other notable applications, they are the fundamental tool in the proof of the [Poincaré conjecture](https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture "Poincaré conjecture") from [geometric topology](https://en.wikipedia.org/wiki/Geometric_topology "Geometric topology").
Partly due to this variety of sources, there is a wide spectrum of types of partial differential equations. Many different methods have been developed for dealing with the individual equations which arise. As such, there is no "universal theory" of partial differential equations, with specialist knowledge being divided between several distinct subfields.[\[2\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-2)
[Ordinary differential equations](https://en.wikipedia.org/wiki/Ordinary_differential_equation "Ordinary differential equation") can be viewed as a subclass of partial differential equations, corresponding to [functions of a single variable](https://en.wikipedia.org/wiki/Function_of_a_real_variable "Function of a real variable"). [Stochastic partial differential equations](https://en.wikipedia.org/wiki/Stochastic_partial_differential_equation "Stochastic partial differential equation") and [nonlocal equations](https://en.wikipedia.org/wiki/Fractional_calculus "Fractional calculus") are widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include [elliptic](https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation "Elliptic partial differential equation") and [parabolic](https://en.wikipedia.org/wiki/Parabolic_partial_differential_equation "Parabolic partial differential equation") partial differential equations, [fluid mechanics](https://en.wikipedia.org/wiki/Fluid_mechanics "Fluid mechanics"), [Boltzmann equations](https://en.wikipedia.org/wiki/Boltzmann_equation "Boltzmann equation"), and [dispersive partial differential equations](https://en.wikipedia.org/wiki/Dispersive_partial_differential_equation "Dispersive partial differential equation").[\[3\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-3)
## Introduction and examples
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=1 "Edit section: Introduction and examples")\]
One of the most important partial differential equations, with many applications, is [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation"). For a function *u*(*x*, *y*, *z*) of three variables, Laplace's equation is  A function that obeys this equation is called a [harmonic function](https://en.wikipedia.org/wiki/Harmonic_function "Harmonic function"). Such functions were widely studied in the 19th century due to their relevance for [classical mechanics](https://en.wikipedia.org/wiki/Classical_mechanics "Classical mechanics"). For example, the equilibrium temperature distribution of a homogeneous solid is a harmonic function. It is usually a matter of straightforward computation to check whether or not a given function is harmonic. For instance   and  are all harmonic, while  is not. It may be surprising that these examples of harmonic functions are of such different forms. This is a reflection of the fact that they are not special cases of a "general solution formula" of Laplace's equation. This is in striking contrast to the case of many [ordinary differential equations](https://en.wikipedia.org/wiki/Ordinary_differential_equation "Ordinary differential equation") (ODEs), where many introductory textbooks aim to find methods leading to general solutions. For Laplace's equation, as for a large number of partial differential equations, such solution formulas do not exist.
This can also be seen in the case of the following PDE: for a function *v*(*x*, *y*) of two variables, consider the equation  It can be directly checked that any function v of the form *v*(*x*, *y*) = *f*(*x*) + *g*(*y*), for any single-variable (differentiable) functions f and g whatsoever, satisfies this condition. This is far beyond the choices available in ODE solution formulas, which typically allow the free choice of some numbers. In the study of PDEs, one generally has the free choice of functions.
The nature of this choice varies from PDE to PDE. To understand it for any given equation, existence and uniqueness theorems are usually important organizational principles. In many introductory textbooks, the role of [existence and uniqueness theorems for ODE](https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem "Picard–Lindelöf theorem") can be somewhat opaque; the existence half is usually unnecessary, since one can directly check any proposed solution formula, while the uniqueness half is often only present in the background to ensure that a proposed solution formula is as general as possible. By contrast, for PDE, existence and uniqueness theorems are often the only means by which one can navigate through the plethora of different solutions at hand. For this reason, they are also fundamental when carrying out a purely numerical simulation, as one must have an understanding of what data is to be prescribed by the user and what is to be left to the computer to calculate.
To discuss such existence and uniqueness theorems, it is necessary to be precise about the [domain](https://en.wikipedia.org/wiki/Domain_of_a_function "Domain of a function") of the "unknown function". Otherwise, speaking only in terms such as "a function of two variables", it is impossible to meaningfully formulate the results. That is, the domain of the unknown function must be regarded as part of the structure of the PDE itself.
The following provides two classic examples of such existence and uniqueness theorems. Even though the two PDEs in question are so similar, there is a striking difference in behavior: for the first PDE, one has the free prescription of a single function, while for the second PDE, one has the free prescription of two functions.
Even more phenomena are possible. For instance, the [following PDE](https://en.wikipedia.org/wiki/Bernstein%27s_problem "Bernstein's problem"), arising naturally in the field of [differential geometry](https://en.wikipedia.org/wiki/Differential_geometry "Differential geometry"), illustrates an example where there is a simple and completely explicit solution formula, but with the free choice of only three numbers and not even one function.
- If u is a function on **R**2 with  then there are numbers a, b, and c with *u*(*x*, *y*) = *ax* + *by* + *c*.
In contrast to the earlier examples, this PDE is **nonlinear**, owing to the square roots and the squares. A **linear** PDE is one such that, if it is homogeneous, the sum of any two solutions is also a solution, and any constant multiple of any solution is also a solution.
A partial differential equation is an equation that involves an unknown function of  variables and (some of) its partial derivatives.[\[4\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-FOOTNOTEEvans19981%E2%80%932-4) That is, for the unknown function  of variables  belonging to the open subset  of , the \-order partial differential equation is defined as ![{\\displaystyle F\[D^{k}u,D^{k-1}u,\\dots ,Du,u,x\]=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6851c09712a689ae776fbfb75da28e8f95f39557) where  and  is the partial derivative operator.
When writing PDEs, it is common to denote partial derivatives using subscripts. For example:  In the general situation that u is a function of n variables, then *u**i* denotes the first partial derivative relative to the i\-th input, *u**ij* denotes the second partial derivative relative to the i\-th and j\-th inputs, and so on.
The Greek letter Δ denotes the [Laplace operator](https://en.wikipedia.org/wiki/Laplace_operator "Laplace operator"); if u is a function of n variables, then  In the physics literature, the Laplace operator is often denoted by ∇2; in the mathematics literature, ∇2*u* may also denote the [Hessian matrix](https://en.wikipedia.org/wiki/Hessian_matrix "Hessian matrix") of u.
### Linear and nonlinear equations
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=5 "Edit section: Linear and nonlinear equations")\]
A PDE is called **linear** if it is linear in the unknown and its derivatives. For example, for a function u of x and y, a second order linear PDE is of the form  where *ai* and *f* are functions of the independent variables x and y only. (Often the mixed-partial derivatives *uxy* and *uyx* will be equated, but this is not required for the discussion of linearity.) If the *ai* are constants (independent of x and y) then the PDE is called **linear with constant coefficients**. If *f* is zero everywhere then the linear PDE is **homogeneous**, otherwise it is **inhomogeneous**. (This is separate from [asymptotic homogenization](https://en.wikipedia.org/wiki/Asymptotic_homogenization "Asymptotic homogenization"), which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.)
Nearest to linear PDEs are **semi-linear** PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order derivatives and the unknown function may appear arbitrarily. For example, a general second order semi-linear PDE in two variables is 
In a **quasilinear** PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives:  Many of the fundamental PDEs in physics are quasilinear, such as the [Einstein equations](https://en.wikipedia.org/wiki/Einstein_equations "Einstein equations") of [general relativity](https://en.wikipedia.org/wiki/General_relativity "General relativity") and the [Navier–Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations "Navier–Stokes equations") describing fluid motion.
A PDE without any linearity properties is called **fully [nonlinear](https://en.wikipedia.org/wiki/Nonlinear_partial_differential_equation "Nonlinear partial differential equation")**, and possesses nonlinearities on one or more of the highest-order derivatives. An example is the [Monge–Ampère equation](https://en.wikipedia.org/wiki/Monge%E2%80%93Amp%C3%A8re_equation "Monge–Ampère equation"), which arises in [differential geometry](https://en.wikipedia.org/wiki/Differential_geometry "Differential geometry").[\[5\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-PrincetonCompanion-5)
### Second order equations
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The elliptic/parabolic/hyperbolic classification provides a guide to appropriate [initial-](https://en.wikipedia.org/wiki/Initial_condition "Initial condition") and [boundary conditions](https://en.wikipedia.org/wiki/Boundary_value_problem "Boundary value problem") and to the [smoothness](https://en.wikipedia.org/wiki/Smoothness "Smoothness") of the solutions. Assuming *uxy* = *uyx*, the general linear second-order PDE in two independent variables has the form  where the coefficients A, B, C... may depend upon x and y. If *A*2 + *B*2 + *C*2 \> 0 over a region of the xy\-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section: 
More precisely, replacing ∂*x* by X, and likewise for other variables (formally this is done by a [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform "Fourier transform")), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a [homogeneous polynomial](https://en.wikipedia.org/wiki/Homogeneous_polynomial "Homogeneous polynomial"), here a [quadratic form](https://en.wikipedia.org/wiki/Quadratic_form "Quadratic form")) being most significant for the classification.
Just as one classifies [conic sections](https://en.wikipedia.org/wiki/Conic_section "Conic section") and quadratic forms into parabolic, hyperbolic, and elliptic based on the [discriminant](https://en.wikipedia.org/wiki/Discriminant "Discriminant") *B*2 − 4*AC*, the same can be done for a second-order PDE at a given point. However, the [discriminant](https://en.wikipedia.org/wiki/Discriminant "Discriminant") in a PDE is given by *B*2 − *AC* due to the convention of the xy term being 2*B* rather than B; formally, the discriminant (of the associated quadratic form) is (2*B*)2 − 4*AC* = 4(*B*2 − *AC*), with the factor of 4 dropped for simplicity.
1. *B*2 − *AC* \< 0 (*[elliptic partial differential equation](https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation "Elliptic partial differential equation")*): Solutions of [elliptic PDEs](https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation "Elliptic partial differential equation") are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation") are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where *x* \< 0. By a change of variables, the equation can always be expressed in the form: where x and y correspond to changed variables. This justifies [Laplace equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation") as an example of this type.[\[6\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-:0-6)
2. *B*2 − *AC* = 0 (*[parabolic partial differential equation](https://en.wikipedia.org/wiki/Parabolic_partial_differential_equation "Parabolic partial differential equation")*): Equations that are [parabolic](https://en.wikipedia.org/wiki/Parabolic_partial_differential_equation "Parabolic partial differential equation") at every point can be transformed into a form analogous to the [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation") by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where *x* = 0. By change of variables, the equation can always be expressed in the form: where x correspond to changed variables. This justifies the [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation"), which is of the form , as an example of this type.[\[6\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-:0-6)
3. *B*2 − *AC* \> 0 (*[hyperbolic partial differential equation](https://en.wikipedia.org/wiki/Hyperbolic_partial_differential_equation "Hyperbolic partial differential equation")*): [hyperbolic](https://en.wikipedia.org/wiki/Hyperbolic_partial_differential_equation "Hyperbolic partial differential equation") equations retain any discontinuities of functions or derivatives in the initial data. An example is the [wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation"). The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where *x* \> 0. By change of variables, the equation can always be expressed in the form: where x and y correspond to changed variables. This justifies the [wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation") as an example of this type.[\[6\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-:0-6)
If there are n independent variables *x*1, *x*2 , …, *x**n*, a general linear partial differential equation of second order has the form 
The classification depends upon the signature of the [eigenvalues](https://en.wikipedia.org/wiki/Eigenvalues "Eigenvalues") of the coefficient matrix *a**i*,*j*.
1. Elliptic: the eigenvalues are all positive or all negative.
2. Parabolic: the eigenvalues are all positive or all negative, except one that is zero.
3. Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
4. Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues.[\[7\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-7)
The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the [Laplace equation](https://en.wikipedia.org/wiki/Laplace_equation "Laplace equation"), the [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation"), and the [wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation").
However, the classification only depends on linearity of the second-order terms and is therefore applicable to semi- and quasilinear PDEs as well. The basic types also extend to hybrids such as the [Euler–Tricomi equation](https://en.wikipedia.org/wiki/Euler%E2%80%93Tricomi_equation "Euler–Tricomi equation"); varying from elliptic to hyperbolic for different [regions](https://en.wikipedia.org/wiki/Region_\(mathematics\) "Region (mathematics)") of the domain, as well as higher-order PDEs, but such knowledge is more specialized.
### Systems of first-order equations and characteristic surfaces
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The classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a [vector](https://en.wikipedia.org/wiki/Euclidean_vector "Euclidean vector") with m components, and the coefficient matrices Aν are m by m matrices for *ν* = 1, 2, …, *n*. The partial differential equation takes the form  where the coefficient matrices Aν and the vector B may depend upon x and u. If a [hypersurface](https://en.wikipedia.org/wiki/Hypersurface "Hypersurface") S is given in the implicit form  where φ has a non-zero gradient, then S is a **characteristic surface** for the [operator](https://en.wikipedia.org/wiki/Differential_operator "Differential operator") L at a given point if the characteristic form vanishes: ![{\\displaystyle Q\\left({\\frac {\\partial \\varphi }{\\partial x\_{1}}},\\ldots ,{\\frac {\\partial \\varphi }{\\partial x\_{n}}}\\right)=\\det \\left\[\\sum \_{\\nu =1}^{n}A\_{\\nu }{\\frac {\\partial \\varphi }{\\partial x\_{\\nu }}}\\right\]=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fae8017409f092e75a9b50bf8e2febcdcdd407c)
The geometric interpretation of this condition is as follows: if data for u are prescribed on the surface S, then it may be possible to determine the normal derivative of u on S from the differential equation. If the data on S and the differential equation determine the normal derivative of u on S, then S is non-characteristic. If the data on S and the differential equation *do not* determine the normal derivative of u on S, then the surface is **characteristic**, and the differential equation restricts the data on S: the differential equation is *internal* to S.
1. A first-order system *Lu* = 0 is *elliptic* if no surface is characteristic for L: the values of u on S and the differential equation always determine the normal derivative of u on S.
2. A first-order system is *hyperbolic* at a point if there is a **spacelike** surface S with normal ξ at that point. This means that, given any non-trivial vector η orthogonal to ξ, and a scalar multiplier λ, the equation *Q*(*λξ* + *η*) = 0 has m real roots *λ*1, *λ*2, …, *λ**m*. The system is **strictly hyperbolic** if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form *Q*(*ζ*) = 0 defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has nm sheets, and the axis *ζ* = *λξ* runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.
## Analytical solutions
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### Separation of variables
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Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a feature of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is *the* solution (this also applies to ODEs). We assume as an [ansatz](https://en.wikipedia.org/wiki/Ansatz "Ansatz") that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.[\[8\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-8)
In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.
This is possible for simple PDEs, which are called [separable partial differential equations](https://en.wikipedia.org/wiki/Separable_partial_differential_equation "Separable partial differential equation"), and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to [diagonal matrices](https://en.wikipedia.org/wiki/Diagonal_matrices "Diagonal matrices") – thinking of "the value for fixed x" as a coordinate, each coordinate can be understood separately.
This generalizes to the [method of characteristics](https://en.wikipedia.org/wiki/Method_of_characteristics "Method of characteristics"), and is also used in [integral transforms](https://en.wikipedia.org/wiki/Integral_transform "Integral transform").
### Method of characteristics
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The characteristic surface in *n* = *2*\-dimensional space is called a **characteristic curve**.[\[9\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-FOOTNOTEZachmanoglouThoe1986115%E2%80%93116-9) In special cases, one can find characteristic curves on which the first-order PDE reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the [method of characteristics](https://en.wikipedia.org/wiki/Method_of_characteristics "Method of characteristics").
More generally, applying the method to first-order PDEs in higher dimensions, one may find characteristic surfaces.
An [integral transform](https://en.wikipedia.org/wiki/Integral_transform "Integral transform") may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.
An important example of this is [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis"), which diagonalizes the heat equation using the [eigenbasis](https://en.wikipedia.org/wiki/Eigenbasis "Eigenbasis") of sinusoidal waves.
If the domain is finite or periodic, an infinite sum of solutions such as a [Fourier series](https://en.wikipedia.org/wiki/Fourier_series "Fourier series") is appropriate, but an integral of solutions such as a [Fourier integral](https://en.wikipedia.org/wiki/Fourier_integral "Fourier integral") is generally required for infinite domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral.
### Change of variables
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=12 "Edit section: Change of variables")\]
Often a PDE can be reduced to a simpler form with a known solution by a suitable [change of variables](https://en.wikipedia.org/wiki/Change_of_variables_\(PDE\) "Change of variables (PDE)"). For example, the [Black–Scholes equation](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation "Black–Scholes equation")  is reducible to the [heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation")  by the change of variables[\[10\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-10) ![{\\displaystyle {\\begin{aligned}V(S,t)&=v(x,\\tau ),\\\\\[5px\]x&=\\ln \\left(S\\right),\\\\\[5px\]\\tau &={\\tfrac {1}{2}}\\sigma ^{2}(T-t),\\\\\[5px\]v(x,\\tau )&=e^{-\\alpha x-\\beta \\tau }u(x,\\tau ).\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e36d50d536184b7e1291e53a595db0b79fad14d9)
### Fundamental solution
\[[edit](https://en.wikipedia.org/w/index.php?title=Partial_differential_equation&action=edit§ion=13 "Edit section: Fundamental solution")\]
Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding the [fundamental solution](https://en.wikipedia.org/wiki/Fundamental_solution "Fundamental solution") (the solution for a point source ), then taking the [convolution](https://en.wikipedia.org/wiki/Convolution "Convolution") with the boundary conditions to get the solution.
This is analogous in [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing") to understanding a filter by its [impulse response](https://en.wikipedia.org/wiki/Impulse_response "Impulse response").
### Superposition principle
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The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example sin *x* + sin *x* = 2 sin *x*. The same principle can be observed in PDEs where the solutions may be real or complex and additive. If *u*1 and *u*2 are solutions of linear PDE in some function space R, then *u* = *c*1*u*1 + *c*2*u*2 with any constants *c*1 and *c*2 are also a solution of that PDE in the same function space.
### Methods for non-linear equations
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There are no generally applicable analytical methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the [Cauchy–Kowalevski theorem](https://en.wikipedia.org/wiki/Cauchy%E2%80%93Kowalevski_theorem "Cauchy–Kowalevski theorem")) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of [analysis](https://en.wikipedia.org/wiki/Mathematical_analysis "Mathematical analysis")).
Nevertheless, some techniques can be used for several types of equations. The [h\-principle](https://en.wikipedia.org/wiki/H-principle "H-principle") is the most powerful method to solve [underdetermined](https://en.wikipedia.org/wiki/Underdetermined_system "Underdetermined system") equations. The [Riquier–Janet theory](https://en.wikipedia.org/w/index.php?title=Riquier%E2%80%93Janet_theory&action=edit&redlink=1 "Riquier–Janet theory (page does not exist)") is an effective method for obtaining information about many analytic [overdetermined](https://en.wikipedia.org/wiki/Overdetermined_system "Overdetermined system") systems.
The [method of characteristics](https://en.wikipedia.org/wiki/Method_of_characteristics "Method of characteristics") can be used in some very special cases to solve nonlinear partial differential equations.[\[11\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-11)
In some cases, a PDE can be solved via [perturbation analysis](https://en.wikipedia.org/wiki/Perturbation_analysis "Perturbation analysis") in which the solution is considered to be a correction to an equation with a known solution. Alternatives are [numerical analysis](https://en.wikipedia.org/wiki/Numerical_analysis "Numerical analysis") techniques from simple [finite difference](https://en.wikipedia.org/wiki/Finite_difference "Finite difference") schemes to the more mature [multigrid](https://en.wikipedia.org/wiki/Multigrid "Multigrid") and [finite element methods](https://en.wikipedia.org/wiki/Finite_element_method "Finite element method"). Many interesting problems in science and engineering are solved in this way using [computers](https://en.wikipedia.org/wiki/Computer "Computer"), sometimes high performance [supercomputers](https://en.wikipedia.org/wiki/Supercomputer "Supercomputer").
From 1870 [Sophus Lie](https://en.wikipedia.org/wiki/Sophus_Lie "Sophus Lie")'s work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called [Lie groups](https://en.wikipedia.org/wiki/Lie_group "Lie group"), be referred, to a common source; and that ordinary differential equations which admit the same [infinitesimal transformations](https://en.wikipedia.org/wiki/Infinitesimal_transformation "Infinitesimal transformation") present comparable difficulties of integration. He also emphasized the subject of [transformations of contact](https://en.wikipedia.org/wiki/Contact_transformation "Contact transformation").
A general approach to solving PDEs uses the symmetry property of differential equations, the continuous [infinitesimal transformations](https://en.wikipedia.org/wiki/Infinitesimal_transformation "Infinitesimal transformation") of solutions to solutions ([Lie theory](https://en.wikipedia.org/wiki/Lie_theory "Lie theory")). Continuous [group theory](https://en.wikipedia.org/wiki/Group_theory "Group theory"), [Lie algebras](https://en.wikipedia.org/wiki/Lie_algebras "Lie algebras") and [differential geometry](https://en.wikipedia.org/wiki/Differential_geometry "Differential geometry") are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its [Lax pairs](https://en.wikipedia.org/wiki/Lax_pair "Lax pair"), recursion operators, [Bäcklund transform](https://en.wikipedia.org/wiki/B%C3%A4cklund_transform "Bäcklund transform") and finally finding exact analytic solutions to the PDE.
Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.
### Semi-analytical methods
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The [Adomian decomposition method](https://en.wikipedia.org/wiki/Adomian_decomposition_method "Adomian decomposition method"),[\[12\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-12) the [Lyapunov](https://en.wikipedia.org/wiki/Aleksandr_Lyapunov "Aleksandr Lyapunov") artificial small parameter method, and his [homotopy perturbation method](https://en.wikipedia.org/wiki/Homotopy_perturbation_method "Homotopy perturbation method") are all special cases of the more general [homotopy analysis method](https://en.wikipedia.org/wiki/Homotopy_analysis_method "Homotopy analysis method").[\[13\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-13) These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known [perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory "Perturbation theory"), thus giving these methods greater flexibility and solution generality.
## Numerical solutions
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The three most widely used [numerical methods to solve PDEs](https://en.wikipedia.org/wiki/Numerical_partial_differential_equations "Numerical partial differential equations") are the [finite element method](https://en.wikipedia.org/wiki/Finite_element_analysis "Finite element analysis") (FEM), [finite volume methods](https://en.wikipedia.org/wiki/Finite_volume_method "Finite volume method") (FVM) and [finite difference methods](https://en.wikipedia.org/wiki/Finite_difference_method "Finite difference method") (FDM), as well other kind of methods called [meshfree methods](https://en.wikipedia.org/wiki/Meshfree_methods "Meshfree methods"), which were made to solve problems where the aforementioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version [hp-FEM](https://en.wikipedia.org/wiki/Hp-FEM "Hp-FEM"). Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM), [extended finite element method](https://en.wikipedia.org/wiki/Extended_finite_element_method "Extended finite element method") (XFEM), [spectral finite element method](https://en.wikipedia.org/wiki/Spectral_element_method "Spectral element method") (SFEM), [meshfree finite element method](https://en.wikipedia.org/wiki/Meshfree_methods "Meshfree methods"), [discontinuous Galerkin finite element method](https://en.wikipedia.org/wiki/Discontinuous_Galerkin_method "Discontinuous Galerkin method") (DGFEM), [element-free Galerkin method](https://en.wikipedia.org/w/index.php?title=Element-free_Galerkin_method&action=edit&redlink=1 "Element-free Galerkin method (page does not exist)") (EFGM), [interpolating element-free Galerkin method](https://en.wikipedia.org/w/index.php?title=Interpolating_element-free_Galerkin_method&action=edit&redlink=1 "Interpolating element-free Galerkin method (page does not exist)") (IEFGM), etc.
### Finite element method
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The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for approximating solutions of partial differential equations (PDE) as well as of integral equations using a finite set of functions.[\[14\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-14)[\[15\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-15) The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.
### Finite difference method
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Finite-difference methods are numerical methods for approximating the solutions to differential equations using [finite difference](https://en.wikipedia.org/wiki/Finite_difference "Finite difference") equations to approximate derivatives.
### Finite volume method
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Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the [divergence theorem](https://en.wikipedia.org/wiki/Divergence_theorem "Divergence theorem"). These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design.
Physics informed neural networks have been used to solve partial differential equations in both forward and inverse problems in a data driven manner.[\[16\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-16) One example is the reconstructing fluid flow governed by the [Navier-Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations "Navier–Stokes equations"). Using physics informed neural networks does not require the often expensive mesh generation that conventional [CFD](https://en.wikipedia.org/wiki/Computational_fluid_dynamics "Computational fluid dynamics") methods rely on.[\[17\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-17)[\[18\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-18) It is evident that geometric and physical constraints have a synergistic effect on neural PDE surrogates, thereby enhancing their efficacy in predicting stable and super long rollouts.[\[19\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-19)
Weak solutions are functions that satisfy the PDE, yet in other meanings than regular sense. The meaning for this term may differ with context, and one of the most commonly used definitions is based on the notion of [distributions](https://en.wikipedia.org/wiki/Schwartz_distribution "Schwartz distribution").
An example[\[20\]](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_note-FOOTNOTEEvans1998chpt._6._Second-Order_Elliptic_Equations-20) for the definition of a weak solution is as follows:
Consider the boundary-value problem given by:  where  denotes a second-order partial differential operator in **divergence form**.
We say a [](https://en.wikipedia.org/wiki/Sobolev_space "Sobolev space") is a weak solution if ![{\\displaystyle \\int \_{U}{\\bigg \[}\\sum \_{i,j}a^{ij}(\\partial \_{i}u)(\\partial \_{j}v)+\\sum \_{i}b^{i}(\\partial \_{i}u)v+cuv{\\bigg \]}dx=\\int \_{U}fvdx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57cca9cc8062c43f270274e8eeba2a5e9ac9dfc8) for every , which can be derived by a formal integral by parts.
An example for a weak solution is as follows:  is a weak solution satisfying  in distributional sense, as formally, 
## Theoretical studies
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In [pure mathematics](https://en.wikipedia.org/wiki/Pure_mathematics "Pure mathematics"), the theoretical studies of PDEs focus on the criteria for a solution to exist and the properties of a solution while finding its formula is often secondary.
Well-posedness refers to a common schematic package of information about a PDE. To say that a PDE is well-posed, one must have:
- an existence and uniqueness theorem, asserting that by the prescription of some freely chosen functions, one can single out one specific solution of the PDE
- by [continuously](https://en.wikipedia.org/wiki/Continuity_\(mathematics\) "Continuity (mathematics)") changing the free choices, one continuously changes the corresponding solution
This is, by the necessity of being applicable to several different PDE, somewhat vague. The requirement of "continuity", in particular, is ambiguous, since there are usually many inequivalent means by which it can be rigorously defined. It is, however, somewhat unusual to study a PDE without specifying a way in which it is well-posed.
Regularity refers to the integrability and differentiability of weak solutions, which can often be represented by [Sobolev spaces](https://en.wikipedia.org/wiki/Sobolev_space "Sobolev space").
This problem arise due to the difficulty in searching for classical solutions. Researchers often tend to find weak solutions at first and then find out whether it is smooth enough to be qualified as a classical solution.
Results from [functional analysis](https://en.wikipedia.org/wiki/Functional_analysis "Functional analysis") are often used in this field of study.
**Some common PDEs**
- [Acoustic wave equation](https://en.wikipedia.org/wiki/Acoustic_wave_equation "Acoustic wave equation")
- [Burgers' equation](https://en.wikipedia.org/wiki/Burgers%27_equation "Burgers' equation")
- [Continuity equation](https://en.wikipedia.org/wiki/Continuity_equation "Continuity equation")
- [Heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation")
- [Helmholtz equation](https://en.wikipedia.org/wiki/Helmholtz_equation "Helmholtz equation")
- [Klein–Gordon equation](https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation "Klein–Gordon equation")
- [Jacobi equation](https://en.wikipedia.org/wiki/Jacobi_equation "Jacobi equation")
- [Lagrange equation](https://en.wikipedia.org/wiki/Lagrange_equation "Lagrange equation")
- [Lorenz equation](https://en.wikipedia.org/wiki/Lorenz_equation "Lorenz equation")
- [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation")
- [Maxwell's equations](https://en.wikipedia.org/wiki/Maxwell%27s_equations "Maxwell's equations")
- [Navier-Stokes equation](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations "Navier–Stokes equations")
- [Poisson's equation](https://en.wikipedia.org/wiki/Poisson%27s_equation "Poisson's equation")
- [Reaction–diffusion system](https://en.wikipedia.org/wiki/Reaction%E2%80%93diffusion_system "Reaction–diffusion system")
- [Schrödinger equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation "Schrödinger equation")
- [Wave equation](https://en.wikipedia.org/wiki/Wave_equation "Wave equation")
**Types of boundary conditions**
- [Dirichlet boundary condition](https://en.wikipedia.org/wiki/Dirichlet_boundary_condition "Dirichlet boundary condition")
- [Neumann boundary condition](https://en.wikipedia.org/wiki/Neumann_boundary_condition "Neumann boundary condition")
- [Robin boundary condition](https://en.wikipedia.org/wiki/Robin_boundary_condition "Robin boundary condition")
- [Cauchy problem](https://en.wikipedia.org/wiki/Cauchy_problem "Cauchy problem")
**Various topics**
- [Jet bundle](https://en.wikipedia.org/wiki/Jet_bundle "Jet bundle")
- [Laplace transform applied to differential equations](https://en.wikipedia.org/wiki/Laplace_transform_applied_to_differential_equations "Laplace transform applied to differential equations")
- [List of dynamical systems and differential equations topics](https://en.wikipedia.org/wiki/List_of_dynamical_systems_and_differential_equations_topics "List of dynamical systems and differential equations topics")
- [Matrix differential equation](https://en.wikipedia.org/wiki/Matrix_differential_equation "Matrix differential equation")
- [Numerical partial differential equations](https://en.wikipedia.org/wiki/Numerical_partial_differential_equations "Numerical partial differential equations")
- [Partial differential algebraic equation](https://en.wikipedia.org/wiki/Partial_differential_algebraic_equation "Partial differential algebraic equation")
- [Recurrence relation](https://en.wikipedia.org/wiki/Recurrence_relation "Recurrence relation")
- [Stochastic processes and boundary value problems](https://en.wikipedia.org/wiki/Stochastic_processes_and_boundary_value_problems "Stochastic processes and boundary value problems")
1. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-1)**
["Regularity and singularities in elliptic PDE's: beyond monotonicity formulas \| EllipticPDE Project \| Fact Sheet \| H2020"](https://cordis.europa.eu/project/id/801867). *CORDIS \| European Commission*. Retrieved 2024-02-05.
2. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-2)**
[Klainerman, Sergiu](https://en.wikipedia.org/wiki/Sergiu_Klainerman "Sergiu Klainerman") (2010). "PDE as a Unified Subject". In Alon, N.; Bourgain, J.; Connes, A.; Gromov, M.; Milman, V. (eds.). *Visions in Mathematics*. Modern Birkhäuser Classics. Basel: Birkhäuser. pp. 279–315\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-0346-0422-2\_10](https://doi.org/10.1007%2F978-3-0346-0422-2_10). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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3. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-3)**
Erdoğan, M. Burak; Tzirakis, Nikolaos (2016). [*Dispersive Partial Differential Equations: Wellposedness and Applications*](https://www.cambridge.org/core/books/dispersive-partial-differential-equations/2DC65286BA080B54EB659E42A553CA88). London Mathematical Society Student Texts. Cambridge: Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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4. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-FOOTNOTEEvans19981%E2%80%932_4-0)** [Evans 1998](https://en.wikipedia.org/wiki/Partial_differential_equation#CITEREFEvans1998), pp. 1–2.
5. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-PrincetonCompanion_5-0)**
Klainerman, Sergiu (2008), "Partial Differential Equations", in Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.), *The Princeton Companion to Mathematics*, Princeton University Press, pp. 455–483
6. ^ [***a***](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-:0_6-0) [***b***](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-:0_6-1) [***c***](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-:0_6-2)
Levandosky, Julie. ["Classification of Second-Order Equations"](https://web.stanford.edu/class/math220a/handouts/secondorder.pdf) (PDF).
7. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-7)** Courant and Hilbert (1962), p.182.
8. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-8)**
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[Adomian, G.](https://en.wikipedia.org/wiki/George_Adomian "George Adomian") (1994). [*Solving Frontier problems of Physics: The decomposition method*](https://books.google.com/books?id=UKPqCAAAQBAJ&q=%22partial+differential%22). Kluwer Academic Publishers. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9789401582896](https://en.wikipedia.org/wiki/Special:BookSources/9789401582896 "Special:BookSources/9789401582896")
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Solin, P. (2005). *Partial Differential Equations and the Finite Element Method*. Hoboken, New Jersey: J. Wiley & Sons. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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Solin, P.; Segeth, K. & Dolezel, I. (2003). *Higher-Order Finite Element Methods*. Boca Raton: Chapman & Hall/CRC Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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18. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-18)**
Raissi, Maziar; Yazdani, Alireza; Karniadakis, George Em (2020-02-28). ["Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7219083). *Science*. **367** (6481): 1026–1030\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2020Sci...367.1026R](https://ui.adsabs.harvard.edu/abs/2020Sci...367.1026R). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1126/science.aaw4741](https://doi.org/10.1126%2Fscience.aaw4741). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [7219083](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7219083). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [32001523](https://pubmed.ncbi.nlm.nih.gov/32001523).
19. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-19)** Huang, Yunfei and Greenberg, David S. "[Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogates](https://arxiv.org/pdf/2506.05513)." Proceedings of the 42th international conference on Machine learning. ACM, 2025.
20. **[^](https://en.wikipedia.org/wiki/Partial_differential_equation#cite_ref-FOOTNOTEEvans1998chpt._6._Second-Order_Elliptic_Equations_20-0)** [Evans 1998](https://en.wikipedia.org/wiki/Partial_differential_equation#CITEREFEvans1998), chpt. 6. Second-Order Elliptic Equations.
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- [Polyanin, A. D.](https://en.wikipedia.org/wiki/Andrei_Polyanin "Andrei Polyanin") (2002), *Handbook of Linear Partial Differential Equations for Engineers and Scientists*, Boca Raton: Chapman & Hall/CRC Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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- [Polyanin, A. D.](https://en.wikipedia.org/wiki/Andrei_Polyanin "Andrei Polyanin") & Zaitsev, V. F. (2004), *Handbook of Nonlinear Partial Differential Equations*, Boca Raton: Chapman & Hall/CRC Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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- Wazwaz, Abdul-Majid (2002). *Partial Differential Equations Methods and Applications*. A.A. Balkema. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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- Zwillinger, D. (1997), *Handbook of Differential Equations* (3rd ed.), Boston: Academic Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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- [Gershenfeld, N.](https://en.wikipedia.org/wiki/Neil_Gershenfeld "Neil Gershenfeld") (1999), *The Nature of Mathematical Modeling* (1st ed.), New York: Cambridge University Press, New York, NY, USA, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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- Krasil'shchik, I.S. & [Vinogradov, A.M., Eds.](https://en.wikipedia.org/wiki/Alexandre_Mikhailovich_Vinogradov "Alexandre Mikhailovich Vinogradov") (1999), *Symmetries and Conservation Laws for Differential Equations of Mathematical Physics*, American Mathematical Society, Providence, Rhode Island, USA, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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`{{citation}}`: CS1 maint: multiple names: authors list ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_multiple_names:_authors_list "Category:CS1 maint: multiple names: authors list")).
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- [Vinogradov, A.M.](https://en.wikipedia.org/wiki/Alexandre_Mikhailovich_Vinogradov "Alexandre Mikhailovich Vinogradov") (2001), *Cohomological Analysis of Partial Differential Equations and Secondary Calculus*, American Mathematical Society, Providence, Rhode Island, USA, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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- [Gustafsson, Bertil](https://en.wikipedia.org/wiki/Bertil_Gustafsson "Bertil Gustafsson") (2008). *High Order Difference Methods for Time Dependent PDE*. Springer Series in Computational Mathematics. Vol. 38. Springer. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/978-3-540-74993-6](https://doi.org/10.1007%2F978-3-540-74993-6). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
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- Zachmanoglou, E. C.; Thoe, Dale W. (1986). *Introduction to Partial Differential Equations with Applications*. New York: Courier Corporation. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-486-65251-3](https://en.wikipedia.org/wiki/Special:BookSources/0-486-65251-3 "Special:BookSources/0-486-65251-3")
.
- [Cajori, Florian](https://en.wikipedia.org/wiki/Florian_Cajori "Florian Cajori") (1928). ["The Early History of Partial Differential Equations and of Partial Differentiation and Integration"](https://web.archive.org/web/20181123102253/http://www.math.harvard.edu/archive/21a_fall_14/exhibits/cajori/cajori.pdf) (PDF). *The American Mathematical Monthly*. **35** (9): 459–467\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2307/2298771](https://doi.org/10.2307%2F2298771). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [2298771](https://www.jstor.org/stable/2298771). Archived from [the original](http://www.math.harvard.edu/archive/21a_fall_14/exhibits/cajori/cajori.pdf) (PDF) on 2018-11-23. Retrieved 2016-05-15.
- [Nirenberg, Louis](https://en.wikipedia.org/wiki/Louis_Nirenberg "Louis Nirenberg") (1994). "Partial differential equations in the first half of the century." Development of mathematics 1900–1950 (Luxembourg, 1992), 479–515, Birkhäuser, Basel.
- [Brezis, Haïm](https://en.wikipedia.org/wiki/Ha%C3%AFm_Brezis "Haïm Brezis"); [Browder, Felix](https://en.wikipedia.org/wiki/Felix_Browder "Felix Browder") (1998). ["Partial Differential Equations in the 20th Century"](https://doi.org/10.1006%2Faima.1997.1713). *[Advances in Mathematics](https://en.wikipedia.org/wiki/Advances_in_Mathematics "Advances in Mathematics")*. **135** (1): 76–144\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1006/aima.1997.1713](https://doi.org/10.1006%2Faima.1997.1713).
- ["Differential equation, partial"](https://www.encyclopediaofmath.org/index.php?title=Differential_equation,_partial), *[Encyclopedia of Mathematics](https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics "Encyclopedia of Mathematics")*, [EMS Press](https://en.wikipedia.org/wiki/European_Mathematical_Society "European Mathematical Society"), 2001 \[1994\]
- [Partial Differential Equations: Exact Solutions](http://eqworld.ipmnet.ru/en/pde-en.htm) at EqWorld: The World of Mathematical Equations.
- [Partial Differential Equations: Index](http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-pde.htm) at EqWorld: The World of Mathematical Equations.
- [Partial Differential Equations: Methods](http://eqworld.ipmnet.ru/en/methods/meth-pde.htm) at EqWorld: The World of Mathematical Equations.
- [Example problems with solutions](https://web.archive.org/web/20170701144823/http://www.exampleproblems.com/wiki/index.php?title=Partial_Differential_Equations) at exampleproblems.com
- [Partial Differential Equations](http://mathworld.wolfram.com/PartialDifferentialEquation.html) at mathworld.wolfram.com
- [Partial Differential Equations](https://reference.wolfram.com/language/FEMDocumentation/tutorial/SolvingPDEwithFEM.html) with Mathematica
- [Partial Differential Equations](http://www.mathworks.com/moler/pdes.pdf) [Archived](https://web.archive.org/web/20160817082230/http://www.mathworks.com/moler/pdes.pdf) 2016-08-17 at the [Wayback Machine](https://en.wikipedia.org/wiki/Wayback_Machine "Wayback Machine") in Cleve Moler: Numerical Computing with MATLAB
- [Partial Differential Equations](http://www.nag.com/numeric/fl/nagdoc_fl24/html/D03/d03intro.html) at nag.com
- Sanderson, Grant (April 21, 2019). ["But what is a partial differential equation?"](https://www.youtube.com/watch?v=ly4S0oi3Yz8&list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6). *3Blue1Brown*. [Archived](https://ghostarchive.org/varchive/youtube/20211102/ly4S0oi3Yz8) from the original on 2021-11-02 – via [YouTube](https://en.wikipedia.org/wiki/YouTube "YouTube").
- [Solutions faibles des équations différentielles](https://hal.science/cel-01196782) at hal.science |
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| Root Hash | 17790707453426894952 |
| Unparsed URL | org,wikipedia!en,/wiki/Partial_differential_equation s443 |