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# 35: Partial differential equations

## Introduction

Like ordinary differential equations, partial differential equations are equations to be solved in which the unknown element is a function, but in PDEs the function is one of several variables, and so of course the known information relates the function and its partial derivatives with respect to the several variables. Again, one generally looks for qualitative statements about the solution. For example, in many cases, solutions exist only if some of the parameters lie in a specific set (say, the set of integers). Various broad families of PDE's admit general statements about the behaviour of their solutions. This area has a long-standing close relationship with the physical sciences, especially physics, thermodynamics, and quantum mechanics: for many of the topics in the field, the origins of the problem and the qualitative nature of the solutions are best understood by describing the corresponding result in physics, as we shall do below.

Roughly corresponding to the initial values in an ODE problem, PDEs are usually solved in the presence of boundary conditions. For example, the Dirichlet problem (actually introduced by Riemann) asks for the solution of the Laplace condition on an open subset D of the plane, with the added condition that the value of u on the boundary of D was to be some prescribed function f. (Physically this corresponds to asking, for example, for the steady-state distribution of electrical charge within D when prescribed voltages are applied around the boundary.) It is a nontrivial task to determine how much boundary information is appropriate for a given PDE!

Linear differential equations occur perhaps most frequently in applications (in settings in which a superposition principle is appropriate.) When these differential equations are first-order, they share many features with ordinary differential equations. (More precisely, they correspond to families of ODEs, in which considerable attention must be focused on the dependence of the solutions on the parameters.)

Historically, three equations were of fundamental interest and exhibit distinctive behaviour. These led to the clarification of three types of second-order linear differential equations of great interest. The Laplace equation

```                             2     2
d u   d u
--- + --- = 0
2     2
dx    dy
```
applies to potential energy functions u=u(x,y) for a conservative force field in the plane. PDEs of this type are called elliptic. The Heat Equation
```                             2     2
d u   d u   d u
--- + --- = ---
2     2   d t
dx    dy
```
applies to the temperature distribution u(x,y) in the plane when heat is allowed to flow from warm areas to cool ones. PDEs of this type are parabolic. The Wave Equation
```                             2     2     2
d u   d u   d u
--- + --- = ---
2     2     2
dx    dy    dt
```
applies to the heights u(x,y) of vibrating membranes and other wave functions. PDEs of this type are called hyperbolic. The analyses of these three types of equations are quite distinct in character. Allowing non-constant coefficients, we see that the solution of a general second-order linear PDE may change character from point to point. These behaviours generalize to nonlinear PDEs as well.

A general linear PDE may be viewed as seeking the kernel of a linear map defined between appropriate function spaces. (Determining which function space is best suited to the problem is itself a nontrivial problem and requires careful functional analysis as well as a consideration of the origin of the equation. Indeed, it is the analysis of PDEs which tends to drive the development of classes of topological vector spaces.) The perspective of differential operators allows the use of general tools from linear algebra, including eigenspace decomposition (spectral theory) and index theory.

Modern approaches seek methods applicable to non-linear PDEs as well as linear ones. In this context existence and uniqueness results, and theorems concerning the regularity of solutions, are more difficult. Since it is unlikely that explicit solutions can be obtained for any but the most special of problems, methods of "solving" the PDEs involve analysis within the appropriate function space -- for example, seeking convergence of a sequence of functions which can be shown to approximately solve the PDE, or describing the sought-for function as a fixed point under a self-map on the function space, or as the point at which some real-valued function is minimized. Some of these approaches may be modified to give algorithms for estimating numerical solutions to a PDE.

Generalizations of results about partial differential equations often lead to statements about function spaces and other topological vector spaces. For example, integral techniques (solving a differential equation by computing a convolution, say) lead to integral operators (transforms on functions spaces); these and differential operators lead in turn to general pseudodifferential operators on function spaces.

## History

See e.g. "Partial Differential Equations in the 20th Century", Haïm Brezis and Felix Browder, Advances in Mathematics 135 (1998) 76-144.

See also Zautykov, O. A.: "Short survey of the development of the theory of partial differential equations. For the 220th anniversary of the appearance of the theory of partial differential equations" (Russian), Vestnik Akad. Nauk Kazah. SSR 1955 (1955), no. 7 (124), 4--19. MR17,931m

## Applications and related fields

Related areas (see diagram below):

• 00, General mathematics
• 22, Topological groups, Lie groups
• 30, Functions of a complex variable
• 31, Potential theory
• 32, Several complex variables and analytic space
• 34, Ordinary differential equations
• 39, Finite differences and functional equations
• 42, Fourier analysis
• 45, Integral equations
• 46, Functional analysis
• 47, Operator theory
• 49, Calculus of variations and optimal control
• 53, Differential geometry
• 58, Global analysis, analysis on manifolds
• 60, Probability theory and stochastic processes
• 65, Numerical analysis
• 73, Mechanics of solids [now 74]
• 76, Fluid mechanics
• 78, Optics, electromagnetic theory
• 80, Classical thermodynamics, heat transfer
• 81, Quantum Theory
• 82, Statistical mechanics, structure of matter
• 83, Relativity and gravitational theory
• 86, Geophysics
• 92, Biology and other natural sciences
• 93, Systems theory; control

For numerical solutions of Partial Differential Equations (including mesh generation) see Numerical Analysis.

This image slightly hand-edited for clarity.

## Subfields

• 35A: General theory
• 35B: Qualitative properties of solutions
• 35C: Representations of solutions
• 35D: Generalized solutions of partial differential equations
• 35F: General first-order equations and systems
• 35G: General higher-order equations and systems
• 35J: Partial differential equations of elliptic type, see also 58GXX, 58G05, 58G10
• 35K: Parabolic equations and systems, see also 35BXX, 35DXX, 35R30, 35R35, 58G11
• 35M: Partial differential equations of special type (mixed, composite, etc.), For degenerate types, see 35J70, 35K65, 35L80
• 35P: Spectral theory and eigenvalue problems for partial differential operators, see also 47AXX, 47BXX, 47F05
• 35Q: Equations of mathematical physics and other areas of application, see also 35J05, 35J10, 35K05, 35L05
• 35R: Miscellaneous topics involving partial differential equations, For equations on manifolds, see 58GXX; for manifolds of solutions, See 58BXX; for stochastic PDEs, See also 60H15
• 35S: Pseudodifferential operators and other generalizations of partial differential operators, see also 47G30, 58G15

This is among the largest areas of the Math Reviews database. The subfield 35Q (applications to physics and other areas) is among the largest three-digit areas as well. Some of the material is categorized in a new area 37: Dynamical systems as of 2000.

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

Winternitz, P.: "What is new in the study of differential equations by group theoretical methods?"; XV International Colloquium on Group Theoretical Methods in Physics (Philadelphia, PA, 1986), 229--248; World Sci. Publishing, Teaneck, NJ, 1987. MR90i:35014

"Reviews in Partial Differential Equations 1980-1986", AMS

Online text on Hilbert space methods [Ralph Showalter]

## Software and tables

Partial differential equations software PDE and Conservation Law software