A mollification regularization method for the inverse spatial-dependent heat source problem

In this paper, we consider the inverse problem of determining a heat source in a parabolic equation where data are given at a fixed time. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. The mollification method with Gauss kernel is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, we also propose a new a posteriori parameter choice rule and get a good error estimate. Numerical results for several benchmark test problems indicate that the method is an accurate and flexible method to determine the unknown spatial-dependent heat source.