Mathematical Methods for Physics and Engineering - Matematica.NET

20.7 UNIQUENESS OF SOLUTIONSEquation type Boundary Conditionshyperbolic open Cauchyparabolic open Dirichlet or Neumannelliptic closed Dirichlet or NeumannTable 20.1 The appropriate boundary conditions for different types of partialdifferential equation.conditions u and ∂u/∂t on the line segment t =0,x =0toL, the solution isspecified in the shaded region.As in the case of first-order PDEs, however, problems can arise. For example,if for a hyperbolic equation the boundary curve intersects any characteristicmore than once then Cauchy conditions along C can overdetermine the problem,resulting in there being no solution. In this case either the boundary curve Cmust be altered, or the boundary conditions on the offending parts of C must berelaxed to Dirichlet or Neumann conditions.The general considerations involved in deciding which boundary conditions areappropriate for a particular problem are complex, and we do not discuss themany further here. § We merely note that whether the various types of boundarycondition are appropriate (in that they give a solution that is unique, sometimesto within a constant, and is well defined) depends upon the type of second-orderequation under consideration and on whether the region of solution is boundedby a closed or an open curve (or a surface if there are more than two independentvariables). Note that part of a closed boundary may be at infinity if conditionsare imposed on u or ∂u/∂n there.It may be shown that the appropriate boundary-condition and equation-typepairings are as given in table 20.1.For example, Laplace’s equation ∇ 2 u = 0 is elliptic and thus requires eitherDirichlet or Neumann boundary conditions on a closed boundary which, as wehave already noted, may be at infinity if the behaviour of u is specified there(most often u or ∂u/∂n → 0 at infinity).20.7 Uniqueness of solutionsAlthough we have merely stated the appropriate boundary types and conditionsfor which, in the general case, a PDE has a unique, well-defined solution, sometimesto within an additive constant, it is often important to be able to provethat a unique solution is obtained.§ For a discussion the reader is referred, for example, to P. M. Morse and H. Feshbach, Methods ofTheoretical Physics, Part I (New York: McGraw-Hill, 1953), chap. 6.705