35: Partial differential equations
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 dyapplies 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 dyapplies 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 dtapplies 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.
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
Related areas (see diagram below):
For numerical solutions of Partial Differential Equations (including mesh generation) see Numerical Analysis.
This image slightly hand-edited for clarity.
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.
Browse all (old) classifications for this area at the AMS.
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]