Elliptic Partial Differential Equations, Relaxation Methods

Heinz Rutishauser

Chapter Chapter 10 in Lectures on Numerical Mathematics, 1990, pp 309-357 from Springer

Abstract: Abstract The classical model examples of partial differential equations are: a) Dirichletproblem (elliptic case): (1) $$\frac{{{\partial ^2}u}}{{\partial {x^2}}}\, + \,\frac{{{\partial ^2}u}}{{\partial {y^2}}}\, = \,f(x,y) in the domain B of the \left( {x,y} \right) - plane,\,$$ u (or ∂u/∂n in the so-called Neumann problem) given on the boundary of B. b) Heat equation (parabolic case): (2) $$\frac{{\partial u}}{{\partial t}} = \frac{{{\partial ^2}u}}{{\partial {x^2}}}for a \leqslant x \leqslant b, t > 0,$$ $$u\left( {x,t} \right) given at t = 0 for all x,$$ $$u or \partial u/\partial x given at x = a,x = b for all t.$$ c) Wave equation (hyperbolic case): (3) $$\frac{{{\partial ^2}u}}{{\partial {t^2}}}{\mkern 1mu} + {\mkern 1mu} \frac{{{\partial ^2}u}}{{\partial {x^2}}}for a \leqslant x \leqslant b, t > 0,$$ $$u and \partial u/\partial t given at t = 0 for all x,$$ $$u or \partial u/\partial x given at x = a, x = b for all t.$$

Date: 1990
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DOI: 10.1007/978-1-4612-3468-5_10

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