Mathematical Methods for Physics and Engineering : A Comprehensive Guide

20.7 UNIQUENESS OF SOLUTIONS

Equation type Boundary Conditions

hyperbolic open Cauchy

parabolic open Dirichlet or Neumann

elliptic closed Dirichlet or Neumann

Table 20.1 The appropriate boundary conditions for different types of partial

differential equation.

conditionsuand∂u/∂ton the line segmentt=0,x=0toL, the solution is

specified 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 characteristic

more than once then Cauchy conditions alongCcan overdetermine the problem,

resulting in there being no solution. In this case either the boundary curveC

must be altered, or the boundary conditions on the offending parts ofCmust be

relaxed to Dirichlet or Neumann conditions.

The general considerations involved in deciding which boundary conditions are

appropriate for a particular problem are complex, and we do not discuss them

any further here.§We merely note that whether the various types of boundary

condition are appropriate (in that they give a solution that is unique, sometimes

to within a constant, and is well defined) depends upon the type of second-order

equation under consideration and on whether the region of solution is bounded

by a closed or an open curve (or a surface if there are more than two independent

variables). Note that part of a closed boundary may be at infinity if conditions

are imposed onuor∂u/∂nthere.

It may be shown that the appropriate boundary-condition and equation-type

pairings are as given in table 20.1.

For example, Laplace’s equation∇^2 u= 0 is elliptic and thus requires either

Dirichlet or Neumann boundary conditions on a closed boundary which, as we

have already noted, may be at infinity if the behaviour ofuis specified there

(most oftenuor∂u/∂n→0 at infinity).

20.7 Uniqueness of solutions

Although we have merely stated the appropriate boundary types and conditions

for which, in the general case, a PDE has a unique, well-defined solution, some-

times to within an additive constant, it is often important to be able to prove

that a unique solution is obtained.

§For a discussion the reader is referred, for example, to P. M. Morse and H. Feshbach,Methods of

Theoretical Physics, Part I(New York: McGraw-Hill, 1953), chap. 6.