A scheme for solving two models of the two-dimensional inverse problem

Inverse problems are of great importance in some engineering texts and many industrial applications. Owing to this, we exhibit a method for numerically estimating two cases of the two-dimensional inverse problems in this research work. The considered inverse problem includes the time-dependent source control parameter r(t). This method is based on operational matrices of differential and integration and product of the shifted Legendre polynomials. Legendre polynomials are implemented to build the homogenizer polynomials. By the use of the method, we reduce the corresponding inverse problem to a system algebraic equations where is easily solvable. It is notable that all the needed computations are done in MATHEMATICATM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{TM}$$\end{document}. Four illustrative examples are applied to investigate the accuracy and applicability of the method.