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[[File:Heat.gif|thumb|right|A visualisation of a solution to the two-dimensional [[Heat equation|heat equation]] with temperature represented by the third dimension]]
In [[Mathematics|mathematics]], a '''partial differential equation''' ('''PDE''') is a [[Differential equation|differential equation]] that contains unknown [[Multivariable calculus|multivariable functions]] and their [[Partial derivative|partial derivative]]s. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is [[Ordinary differential equation|ordinary differential equation]]s (ODEs), which deal with functions of a single variable and their derivatives.

PDEs can be used to describe a wide variety of phenomena such as sound, heat, [[Physics:Diffusion|diffusion]], [[Physics:Electrostatics|electrostatics]], [[Physics:Electromagnetism|electrodynamics]], [[Physics:Fluid dynamics|fluid dynamics]], [[Physics:Elasticity|elasticity]], gravitation and [[Physics:Quantum mechanics|quantum mechanics]]. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional [[Dynamical system|dynamical system]]s, partial differential equations often model [[Multidimensional system|multidimensional system]]s. PDEs find their generalisation in [[Stochastic partial differential equation|stochastic partial differential equation]]s.

== Introduction ==
Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. For example, the position of a [[Rigid body|rigid body]] is specified by six parameters,<ref>{{Cite book|url=https://books.google.com/books?id=v9PLbcYd9aUC&pg=PA32|title=Modelling and Control of Robot Manipulators|last=Sciavicco|first=Lorenzo|last2=Siciliano|first2=Bruno|date=2001-02-19|publisher=Springer Science & Business Media|isbn=9781852332211|language=en}}</ref> but the configuration of a [[Physics:Fluid|fluid]] is given by the continuous distribution of several parameters, such as the [[Physics:Temperature|temperature]], [[Physics:Pressure|pressure]], and so forth. The dynamics for the rigid body take place in a finite-dimensional [[Physics:Configuration space|configuration space]]; the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than ordinary differential equations (ODEs), but here again, there will be simple solutions for linear problems. Classic domains where PDEs are used include [[Physics:Acoustics|acoustics]], [[Physics:Fluid dynamics|fluid dynamics]], [[Physics:Electrodynamics|electrodynamics]], and [[Physics:Heat transfer|heat transfer]].

A partial differential equation (PDE) for the function {{math|''u''(''x''<sub>1</sub>,… ''x<sub>n</sub>'')}} is an equation of the form

: <math>f \left (x_1, \ldots x_n; u, \frac{\partial u}{\partial x_1}, \ldots \frac{\partial u}{\partial x_n}; \frac{\partial^2 u}{\partial x_1 \partial x_1}, \ldots \frac{\partial^2 u}{\partial x_1 \partial x_n}; \ldots \right) = 0.</math>

If {{mvar|f}} is a [[Linear function|linear function]] of {{mvar|u}} and its derivatives, then the PDE is called linear. Common examples of linear PDEs include the [[Heat equation|heat equation]], the [[Physics:Wave equation|wave equation]], [[Laplace's equation]], [[Helmholtz equation]], [[Physics:Klein–Gordon equation|Klein–Gordon equation]], and [[Physics:Poisson's equation|Poisson's equation]].

A relatively simple PDE is

: <math>\frac{\partial u}{\partial x}(x,y) = 0.</math>

This relation implies that the function {{math|''u''(''x'',''y'')}} is independent of {{mvar|x}}. However, the equation gives no information on the function's dependence on the variable {{mvar|y}}. Hence the general solution of this equation is

: <math>u(x,y) = f(y),</math>

where {{mvar|f}} is an arbitrary function of {{mvar|y}}. The analogous ordinary differential equation is

: <math>\frac{\mathrm{d} u}{\mathrm{d} x}(x) = 0,</math>

which has the solution

: <math>u(x) = c,</math>

where {{mvar|c}} is any [[Constant (mathematics)|constant]] value. These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions.

A solution of a PDE is generally not [[Uniqueness quantification|unique]]; additional conditions must generally be specified on the [[Boundary (topology)|boundary]] of the region where the solution is defined. For instance, in the simple example above, the function {{math|''f''(''y'')}} can be determined if {{mvar|u}} is specified on the line {{math|''x'' {{=}} 0}}.

== Existence and uniqueness ==
Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the [[Picard–Lindelöf theorem]], that is far from the case for partial differential equations. The [[Cauchy–Kowalevski theorem]] states that the [[Cauchy problem]] for any partial differential equation whose coefficients are [[Analytic function|analytic]] in the unknown function and its derivatives, has a locally unique analytic solution. Although this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: see [[Lewy's example|Lewy (1957)]]. Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties.  The mathematical study of these questions is usually in the more powerful context of [[Weak solution|weak solution]]s.

An example of pathological behavior is the sequence (depending upon {{mvar|n}}) of [[Cauchy problem]]s for the Laplace equation

: <math>\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0,</math>

with boundary conditions

: <math>\begin{align} u(x,0) &= 0, \\ \frac{\partial u}{\partial y}(x,0) &= \frac{\sin nx}{n}, \end{align}</math>

where {{mvar|n}} is an integer. The derivative of {{mvar|u}} with respect to {{mvar|y}} approaches zero [[Uniform convergence|uniformly]] in {{mvar|x}} as {{mvar|n}} increases, but the solution is

: <math>u(x,y) = \frac{\sinh ny \sin nx}{n^2}.</math>

This solution approaches infinity if {{mvar|nx}} is not an integer multiple of {{pi}} for any non-zero value of {{mvar|y}}. The Cauchy problem for the Laplace equation is called ''ill-posed'' or ''not [[Well-posed problem|well-posed]]'', since the solution does not continuously depend on the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications.

The [[Navier–Stokes existence and smoothness|existence of solutions for the Navier–Stokes equations]], a partial differential equation used in [[Physics:Fluid mechanics|fluid mechanics]], is part of one of the Millennium Prize Problems.

== Notation ==
In PDEs, it is common to denote partial derivatives using subscripts. That is:

: <math>u_x = \frac{\partial u}{\partial x}</math>
: <math>u_{xx} = \frac{\partial^2 u}{\partial x^2} </math>
: <math>u_{xy} = \frac{\partial^2 u}{\partial y\, \partial x} = \frac{\partial}{\partial y } \left(\frac{\partial u}{\partial x}\right). </math>

Especially in physics, [[Del|del]] or nabla ({{math|∇}}) is often used to denote spatial derivatives, and {{math|''&#x307;u'', ''ü''}} for time derivatives. For example, the [[Physics:Wave equation|wave equation]] (mentioned below) can be written as

:<math>\ddot u=c^2\nabla^2u</math>

or

:<math>\ddot u=c^2\Delta u</math>

where {{math|Δ}} is the [[Laplace operator]].

== Classification ==
Some linear, second-order partial differential equations can be classified as [[Parabolic partial differential equation|parabolic]], [[Hyperbolic partial differential equation|hyperbolic]] and [[Elliptic partial differential equation|elliptic]]. Others, such as the [[Euler–Tricomi equation]], have different types in different regions. The classification provides a guide to appropriate initial and [[Boundary value problem|boundary conditions]] and to the smoothness of the solutions.

=== Equations of first order ===

{{main|First-order partial differential equation}}

=== Linear equations of second order ===
Assuming {{math|''u<sub>xy</sub>'' {{=}} ''u<sub>yx</sub>''}}, the general linear second-order PDE in two independent variables has the form

: <math>Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots \mbox{(lower order terms)} = 0,</math>

where the coefficients {{mvar|A}}, {{mvar|B}}, {{mvar|C}}... may depend upon {{mvar|x}} and {{mvar|y}}. If {{math|''A''<sup>2</sup> + ''B''<sup>2</sup> + ''C''<sup>2</sup> > 0}} over a region of the {{mvar|xy}}-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:

: <math>Ax^2 + 2Bxy + Cy^2 + \cdots = 0.</math>
More precisely, replacing {{math|∂<sub>''x''</sub>}} by {{mvar|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|homogeneous polynomial]], here a [[Quadratic form|quadratic form]]) being most significant for the classification.

Just as one classifies [[Conic section|conic section]]s and quadratic forms into parabolic, hyperbolic, and elliptic based on the [[Discriminant|discriminant]] {{math|''B''<sup>2</sup> − 4''AC''}}, the same can be done for a second-order PDE at a given point.  However, the [[Discriminant|discriminant]] in a PDE is given by {{math|''B''<sup>2</sup> − ''AC''}} due to the convention of the {{mvar|xy}} term being {{math|2''B''}} rather than {{mvar|B}}; formally, the discriminant (of the associated quadratic form) is {{math|(2''B'')<sup>2</sup> − 4''AC'' {{=}} 4(''B''<sup>2</sup> − ''AC'')}}, with the factor of 4 dropped for simplicity.

# {{math|''B''<sup>2</sup> − ''AC'' < 0}} (''[[Elliptic partial differential equation|elliptic partial differential equation]]''): Solutions of [[Elliptic partial differential equation|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 {{math|''x'' < 0}}.
# {{math|''B''<sup>2</sup> − ''AC'' {{=}} 0}} (''[[Parabolic partial differential equation|parabolic partial differential equation]]''): Equations that are [[Parabolic partial differential equation|parabolic]] at every point can be transformed into a form analogous to the [[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 {{math|''x'' {{=}} 0}}.
# {{math|''B''<sup>2</sup> − ''AC'' > 0}} (''[[Hyperbolic partial differential equation|hyperbolic partial differential equation]]''): [[Hyperbolic partial differential equation|hyperbolic]] equations retain any discontinuities of functions or derivatives in the initial data. An example is the [[Physics: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 {{math|''x'' > 0}}.

If there are {{mvar|n}} independent variables {{math|''x''<sub>1</sub>, ''x''<sub>2 </sub>,… ''x''<sub>''n''</sub>}}, a general linear partial differential equation of second order has the form

: <math>L u =\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\partial^2 u}{\partial x_i \partial x_j} \quad \text{ plus lower-order terms} =0.</math>

The classification depends upon the signature of the eigenvalues of the coefficient matrix {{math|''a''<sub>''i'',''j''</sub>}}.

# Elliptic: the eigenvalues are all positive or all negative.
# Parabolic: the eigenvalues are all positive or all negative, save 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. There is only a limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).

=== Systems of first-order equations and characteristic surfaces ===
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown {{mvar|u}} is now a [[Euclidean vector|vector]] with {{mvar|m}} components, and the coefficient matrices {{mvar|A<sub>ν</sub>}} are {{mvar|m}} by {{mvar|m}} matrices for {{math|''ν'' {{=}} 1, 2,… ''n''}}. The partial differential equation takes the form

: <math>Lu = \sum_{\nu=1}^{n} A_\nu \frac{\partial u}{\partial x_\nu} + B=0,</math>

where the coefficient matrices {{mvar|A<sub>ν</sub>}} and the vector {{mvar|B}} may depend upon {{mvar|x}} and {{mvar|u}}. If a [[Hypersurface|hypersurface]] {{mvar|S}} is given in the implicit form

: <math>\varphi(x_1, x_2, \ldots x_n)=0,</math>

where {{mvar|φ}} has a non-zero gradient, then {{mvar|S}} is a '''characteristic surface''' for the operator {{mvar|L}} at a given point if the characteristic form vanishes:

: <math>Q\left(\frac{\partial\varphi}{\partial x_1}, \ldots\frac{\partial\varphi}{\partial x_n}\right) =\det\left[\sum_{\nu=1}^nA_\nu \frac{\partial \varphi}{\partial x_\nu}\right]=0.\,</math>

The geometric interpretation of this condition is as follows: if data for {{mvar|u}} are prescribed on the surface {{mvar|S}}, then it may be possible to determine the normal derivative of {{mvar|u}} on {{mvar|S}} from the differential equation. If the data on {{mvar|S}} and the differential equation determine the normal derivative of {{mvar|u}} on {{mvar|S}}, then {{mvar|S}} is non-characteristic. If the data on {{mvar|S}} and the differential equation ''do not'' determine the normal derivative of {{mvar|u}} on {{mvar|S}}, then the surface is '''characteristic''', and the differential equation restricts the data on {{mvar|S}}: the differential equation is ''internal'' to {{mvar|S}}.

# A first-order system {{math|''Lu'' {{=}} 0}} is ''elliptic'' if no surface is characteristic for {{mvar|L}}: the values of {{mvar|u}} on {{mvar|S}} and the differential equation always determine the normal derivative of {{mvar|u}} on {{mvar|S}}.
# A first-order system is ''hyperbolic'' at a point if there is a '''spacelike''' surface {{mvar|S}} with normal {{mvar|ξ}} at that point. This means that, given any non-trivial vector {{mvar|η}} orthogonal to {{mvar|ξ}}, and a scalar multiplier {{mvar|λ}}, the equation {{math|''Q''(''λξ'' + ''η'') {{=}} 0}} has {{mvar|m}} real roots {{math|''λ''<sub>1</sub>, ''λ''<sub>2</sub>,… ''λ''<sub>''m''</sub>}}. The system is '''strictly hyperbolic''' if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form {{math|''Q''(''ζ'') {{=}} 0}} defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has {{mvar|m}} sheets, and the axis {{math|''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.

=== Equations of mixed type ===
If a PDE has coefficients that are not constant, it is possible that it will not belong to any of these categories but rather be of '''mixed type'''. A simple but important example is the [[Euler–Tricomi equation]]

: <math>u_{xx} = xu_{yy},</math>

which is called '''elliptic-hyperbolic''' because it is elliptic in the region {{math|''x'' < 0}}, hyperbolic in the region {{math|''x'' > 0}}, and degenerate parabolic on the line {{math|''x'' {{=}} 0}}.
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=== Infinite-order PDEs in quantum mechanics ===
In the phase space formulation of quantum mechanics,  one may consider the [[Method of quantum characteristics|quantum Hamilton's equations]] for trajectories of quantum particles. These equations are infinite-order PDEs. However, in the semiclassical expansion, one has a finite system of ODEs at any fixed order of {{mvar|ħ}}.  The evolution equation of the Wigner function is also an infinite-order PDE. The quantum trajectories are [[Method of quantum characteristics|quantum characteristics]], with the use of which one could calculate the evolution of the Wigner function.

== Analytical solutions ==

===Separation of variables===
{{main|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 characteristic 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 [[Physics: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.<ref>{{cite book|last1=Gershenfeld|first1=Neil|title=The nature of mathematical modeling|url=https://archive.org/details/naturemathematic00gers_334|url-access=limited|date=2000|publisher=Cambridge Univ. Press|location=Cambridge|isbn=0521570956|page=[https://archive.org/details/naturemathematic00gers_334/page/n32 27]|edition=Reprinted (with corr.)}}</ref>

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 equation|separable partial differential equation]]s, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to diagonal matrices – thinking of "the value for fixed {{mvar|x}}" as a coordinate, each coordinate can be understood separately.

This generalizes to the [[Method of characteristics|method of characteristics]], and is also used in [[Integral transform|integral transform]]s.

===Method of characteristics===
{{main|Method of characteristics}}
In special cases, one can find characteristic curves on which the equation reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the [[Method of characteristics|method of characteristics]].

More generally, one may find characteristic surfaces.

===Integral transform===
An [[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]], 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===

Often a PDE can be reduced to a simpler form with a known solution by a suitable [[Change of variables (PDE)|change of variables]].  For example, the [[Finance:Black–Scholes equation#Derivation|Black–Scholes]] PDE

:<math> \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 </math>

is reducible to the [[Heat equation|heat equation]]

:<math> \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2}</math>

by the change of variables (for complete details see {{webarchive |url=https://web.archive.org/web/20080411030405/http://www.math.unl.edu/~sdunbar1/Teaching/MathematicalFinance/Lessons/BlackScholes/Solution/solution.shtml |date=April 11, 2008 |title=Solution of the Black Scholes Equation }})

:<math>\begin{align}
V(S,t) &= K v(x,\tau),\\[5px]
x &= \ln\left(\tfrac{S}{K} \right),\\[5px]
\tau &= \tfrac{1}{2} \sigma^2 (T - t),\\[5px]
v(x,\tau)&=e^{-\alpha x-\beta\tau} u(x,\tau).
\end{align}</math>

===Fundamental solution===
{{main|Fundamental solution}}
Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding the [[Fundamental solution|fundamental solution]] (the solution for a point source), then taking the [[Convolution|convolution]] with the boundary conditions to get the solution.

This is analogous in [[Signal processing|signal processing]] to understanding a filter by its [[Impulse response|impulse response]].

===Superposition principle===
{{further| 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 {{math|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. [[Superposition principle|superposition]]
If {{math|''u''<sub>1</sub>}} and {{math|''u''<sub>2</sub>}} are solutions of linear PDE in some function space {{mvar|R}}, then {{math|''u'' {{=}} ''c''<sub>1</sub>''u''<sub>1</sub> + ''c''<sub>2</sub>''u''<sub>2</sub>}} with any constants {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}} are also a solution of that PDE in the same function space.

===Methods for non-linear equations===

There are no generally applicable 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 [[Mathematical analysis|analysis]]). Computational solution to the nonlinear PDEs, the [[Split-step method|split-step method]], exist for specific equations like [[Nonlinear Schrödinger equation|nonlinear Schrödinger equation]].

Nevertheless, some techniques can be used for several types of equations. The {{mvar|h}}-principle is the most powerful method to solve [[Underdetermined system|underdetermined]] equations. The Riquier–Janet theory is an effective method for obtaining information about many analytic [[Overdetermined system|overdetermined]] systems.

The [[Method of characteristics|method of characteristics]] can be used in some very special cases to solve partial differential equations.

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|numerical analysis]] techniques from simple [[Finite difference|finite difference]] schemes to the more mature multigrid and [[Finite element method|finite element method]]s. Many interesting problems in science and engineering are solved in this way using [[Computer|computer]]s, sometimes high performance [[Supercomputer|supercomputer]]s.

===Lie group method===
From 1870 [[Biography: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 group]]s, be referred, to a common source; and that ordinary differential equations which admit the same [[Infinitesimal transformation|infinitesimal transformation]]s 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 transformation|infinitesimal transformation]]s of solutions to solutions ([[Lie theory]]). Continuous [[Group theory|group theory]], Lie algebras and [[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 pair]]s, 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.

===Semianalytical methods===
The [[Adomian decomposition method]], the [[Biography:Aleksandr Lyapunov|Lyapunov]] artificial small parameter method, and He's homotopy perturbation method are all special cases of the more general [[Homotopy analysis method|homotopy analysis method]]. These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known [[Physics:Perturbation theory|perturbation theory]], thus giving these methods greater flexibility and solution generality.

== Numerical solutions ==
The three most widely used [[Numerical partial differential equations|numerical methods to solve PDEs]] are the finite element method (FEM), [[Finite volume method|finite volume method]]s (FVM) and [[Finite difference method|finite difference method]]s (FDM), as well other kind of methods called [[Meshfree methods]], which were made to solve problems where the before mentioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version [[Hp-FEM|hp-FEM]]. Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM), [[Physics:Extended finite element method|extended finite element method]] (XFEM), [[Spectral element method|spectral finite element method]] (SFEM), [[Meshfree methods|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 ===
{{main|Finite element method}}
The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. 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===
{{main|Finite difference method}}
Finite-difference methods are numerical methods for approximating the solutions to differential equations using [[Finite difference|finite difference]] equations to approximate derivatives.

=== Finite volume method ===
{{main|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|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.

==See also==
* [[Dirichlet boundary condition]]
* [[Jet bundle]]
* [[Laplace transform applied to differential equations]]
* List of dynamical systems and differential equations topics
* [[Matrix differential equation]]
* [[Neumann boundary condition]]
* [[Numerical partial differential equations]]
* [[Partial differential algebraic equation]]
* [[Recurrence relation]]
* [[Robin boundary condition]]
* [[Stochastic processes and boundary value problems]]

== The energy method ==
The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems.<ref>{{cite book |title=High Order Difference Methods for Time Dependent PDE
|first=Bertil|last=Gustafsson|publisher=Springer|year=2008|isbn=978-3-540-74992-9|doi=10.1007/978-3-540-74993-6}}</ref> In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by

: <math>\frac{\partial u}{\partial t} + \alpha \frac{\partial u}{\partial x} = 0, \quad x \in [a,b], \operatorname t > 0,</math>

where <math>\alpha \neq 0</math> is a constant and <math>u(x,t)</math> is an unknown function with initial condition <math>u(x,0) = f(x)</math>. Multiplying with <math>u</math> and integrating over the domain gives

: <math>\int_a^b u \frac{\partial u}{\partial t} \operatorname dx + \alpha \int _a ^b u \frac{\partial u}{\partial x} \operatorname dx = 0.</math>

Using that

: <math>\int _a ^b u \frac{\partial u}{\partial t} \operatorname dx = \frac{1}{2} \frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \quad \text{and} \quad \int _a ^b u \frac{\partial u}{\partial x} \operatorname dx = \frac{1}{2} u(b,t)^2 - \frac{1}{2} u(a,t)^2,.</math>
where integration by parts has been used for the second relationship, we get

: <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 + \alpha u(b,t)^2 - \alpha u(a,t)^2 = 0.</math>

Here <math>\vert \vert \cdot \vert \vert</math> denotes the standard L2-norm.
For well-posedness we require that the energy of the solution is non-increasing, i.e. that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>, which is achieved by specifying <math>u</math> at <math>x = a</math> if <math>\alpha > 0</math> and at <math>x = b</math> if <math>\alpha < 0</math>. This corresponds to only imposing boundary conditions at the inflow. Note that well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> holds when all data is set to zero.

==Notes==
{{Reflist}}

== References ==
{{refbegin|2}}
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{{refend}}

== Further reading ==
* {{Cite journal|last=Cajori|first=Florian|year=1928|title=The Early History of Partial Differential Equations and of Partial Differentiation and Integration|url=http://www.math.harvard.edu/archive/21a_fall_14/exhibits/cajori/cajori.pdf|journal=The American Mathematical Monthly|volume=35|issue=9|pages=459–467|doi=10.2307/2298771|access-date=2016-05-15|archive-url=https://web.archive.org/web/20181123102253/http://www.math.harvard.edu/archive/21a_fall_14/exhibits/cajori/cajori.pdf|archive-date=2018-11-23|url-status=dead}}

== External links ==
* {{springer|title=Differential equation, partial|id=p/d031920}}
* [http://eqworld.ipmnet.ru/en/pde-en.htm Partial Differential Equations: Exact Solutions] at EqWorld: The World of Mathematical Equations.
* [http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-pde.htm Partial Differential Equations: Index] at EqWorld: The World of Mathematical Equations.
* [http://eqworld.ipmnet.ru/en/methods/meth-pde.htm Partial Differential Equations: Methods] at EqWorld: The World of Mathematical Equations.
* [https://web.archive.org/web/20170701144823/http://www.exampleproblems.com/wiki/index.php?title=Partial_Differential_Equations Example problems with solutions] at exampleproblems.com
* [http://mathworld.wolfram.com/PartialDifferentialEquation.html Partial Differential Equations] at mathworld.wolfram.com
* [https://reference.wolfram.com/language/FEMDocumentation/tutorial/SolvingPDEwithFEM.html Partial Differential Equations] with Mathematica
* [http://www.mathworks.com/moler/pdes.pdf Partial Differential Equations] in Cleve Moler: Numerical Computing with MATLAB
* [http://www.nag.com/numeric/fl/nagdoc_fl24/html/D03/d03intro.html Partial Differential Equations] at nag.com
* {{cite web |first=Grant |last=Sanderson |title=But what is a partial differential equation? |work=3Blue1Brown |date=April 21, 2019 |url=https://www.youtube.com/watch?v=ly4S0oi3Yz8&list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6 |via=[[Company:YouTube|YouTube]] }}

{{Differential equations topics}}

[[Category:Partial differential equations| ]]
[[Category:Multivariable calculus]]
[[Category:Differential equations]]
[[Category:Concepts in physics]]

{{Sourceattribution|Partial differential equation}}