An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation

In this article, simultaneous identification of the timewise lowest and source terms in a two-dimensional (2D) parabolic equation from knowledge of additional measurements is studied. Existence and uniqueness of the solution is proved by means of the contraction mapping on a small time interval. Since the problem is yet ill- posed (very small errors in the time-average temperature input may cause large errors in the output source and potential functions), we need to regularize the solution. Hence, regularization is needed for the retrieval of unknown terms. The 2D equation is computationally solved using the ADE scheme and reshaped as non-linear optimization of the Tikhonov regularization function. This is numerically studied by means of the MATLAB $lsqnonlin$ subroutine. Finally, we present a numerical example to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the ADE is an efficient and unconditionally stable scheme for reconstructing the potential and source coefficients from minimal data which makes the solution of the inverse problem (IP) unique.