NUMERICAL-SOLUTION OF THE SIDEWAYS HEAT-EQUATION BY DIFFERENCE APPROXIMATION IN TIME

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 sideways heat eguation is a model of a problem where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. The heat equation is discretized by a differential-difference equation, where the time derivative has been replaced by a finite difference, and we analyse the approximation properties of time-discrete approximations using Fourier transform techniques. An error estimate is obtained for one such approximation, and it is shown that when the data error tends to zero, the error in the approximate solution tends to zero logarithmically. This error estimate also gives information about how to choose the step length in the time discretization.