Solving an Inverse Heat Conduction Problem by a 'Method of Lines'

Lars Eld\'en
Department of Mathematics, Link\"oping University
 S-581 83 Link\"oping
Sweden


ABSTRACT

We consider a Cauchy problem for the heat equation in the quarter
plane, where data are given at $x=1$ and a solution is sought in the
interval $0 < x < 1$. This {\sl inverse heat conduction problem} is a
model of a situation, where one wants to determine the surface
temperature, given measurements inside a heat-conducting body.  The
problem is ill--posed in the sense that the solution (if it exists)
does not depend continuously on the data.  In an earlier paper we
showed that replacement of the time derivative by a difference
stabilizes the problem. In this paper we investigate the use of time
differencing combined with a 'method of lines' for solving numerically
the initial value problem in the space variable. We discuss the
numerical stability of this procedure, and show that in most cases a
usual explicit (e.g. Runge-Kutta) method, can be used efficiently and
stably. Numerical examples are given. The approach of this paper is
proposed as an alternative way of implementing space-marching methods
for the sideways heat equation.