Elliptic Partial Differential Equations, Relaxation Methods

Abstract

The classical model examples of partial differential equations are:

• a) Dirichletproblem (elliptic case):

  $$\frac{{{\partial ^2}u}}{{\partial {x^2}}}\, + \,\frac{{{\partial ^2}u}}{{\partial {y^2}}}\, = \,f(x,y) in the domain B of the \left( {x,y} \right) - plane,\,$$

  ((1))

  u (or ∂u/∂n in the so-called Neumann problem) given on the boundary of B.

• b) Heat equation (parabolic case):

  $$\frac{{\partial u}}{{\partial t}} = \frac{{{\partial ^2}u}}{{\partial {x^2}}}for a \leqslant x \leqslant b, t > 0,$$

  ((2))
  $$u\left( {x,t} \right) given at t = 0 for all x,$$
  $$u or \partial u/\partial x given at x = a,x = b for all t.$$

• c) Wave equation (hyperbolic case):

  $$\frac{{{\partial ^2}u}}{{\partial {t^2}}}{\mkern 1mu} + {\mkern 1mu} \frac{{{\partial ^2}u}}{{\partial {x^2}}}for a \leqslant x \leqslant b, t > 0,$$

  ((3))
  $$u and \partial u/\partial t given at t = 0 for all x,$$
  $$u or \partial u/\partial x given at x = a, x = b for all t.$$