Simultaneous determination of source term and initial value in the heat conduction problem by modified quasi-reversibility regularization method

Abstract

In this article, we investigate the inverse problem of determining source term and initial data simultaneously in a parabolic equation where data are given at two fixed times. We can get the exact solution of the problem by the Fourier transform, and find that the problem is ill-posed, i.e., the solution does not depend continuously on the input data and small errors in the data can destroy the numerical solution. Hence, a modified quasi-reversibility regularization method is used to solve the problem and an order-optimal error estimate is obtained under a suitable parameter choice rule. Finally, some numerical examples including smooth and nonsmooth functions show that the proposed method is effective and stable.