Determination of an Unknown Time-dependent Heat Source from A Nonlocal Measurement by Finite Difference Method

In this paper, we consider an inverse time-dependent source problem of heat conduction equation. Firstly, the ill-posedness and conditional stability of this inverse source problem is analyzed. Then, a finite difference inversion method is proposed for reconstructing the time-dependent source from a nonlocal measurement. The existence and uniqueness of the finite difference inverse solutions are rigorously analyzed, and the convergence is proved. Combined with the mollification method, the proposed finite difference inversion method can obtain more stable reconstructions from the nonlocal data with noise. Finally, numerical examples are given to illustrate the efficiency and convergence of the proposed finite difference inversion method.