Recovering Time-Dependent Coefficients in a Two-Dimensional Parabolic Equation Using Nonlocal Overspecified Conditions via ADE Finite Difference Schemes

Abstract

This study delves into the nonlocal inverse boundary-value problem for a second-order, two-dimensional parabolic equation within a rectangular domain. The primary focus is to identify the unknown coefficient and propose a resolution to the problem. The second-order, two-dimensional convection equation is addressed through the direct application of the alternating direction explicit (ADE) finite difference scheme. An adaptation of the ADE scheme is formulated to accommodate mixed boundary conditions, utilizing suitable expressions at the boundaries. Furthermore, unconditional stability is scrutinized through a series of examples. Each ADE scheme typically comprises two substeps, known as upward and downward sweeps, during which values computed at the new time level are incorporated into the discretization template. The inverse problem is restructured into a nonlinear regularized least-square optimization problem, with a defined boundary for the unknown factor, and is effectively resolved using the MATLAB subroutine lsqnonlin from the optimization toolbox. Given the typically ill-posed nature of the problem under investigation, where minor errors in the input data can significantly affect the output, Tikhonov's regularization technique is employed to produce stable and regularized results.