Numerical solution of a source identification problem by Fourier method

Abstract

A source identification problem for heat transfer equation in the case of given final observation data is considered. The source density has the form f(x)g(t), where the function g(t) is given, and the function f(x) is to be determined. The necessary condition of the minimum of the residual functional, which of consists of related direct and adjoint problem, is written. The analytical representation of the solution in the form of Fourier series is obtained. Then the series are calculated numerically. In several cases depending on the behavior of the function g(t) a recovery of the unknown solution have been obtained with machine accuracy. Discontinuous and oscillating functions are recovered with sufficiently accuracy as well.