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 MATHEMATICA\(^{TM}\). Four illustrative examples are applied to investigate the accuracy and applicability of the method.

反问题在某些工程文本和许多工业应用中非常重要。因此，在本研究工作中，我们展示了一种用于数值估计二维逆问题的两种情况的方法。所考虑的逆问题包括与时间有关的源控制参数r（t）。该方法基于微分和积分的运算矩阵以及移位的勒让德多项式的乘积。实施勒让德多项式以构建均化器多项式。通过使用该方法，我们将相应的反问题简化为易于求解的系统代数方程。值得注意的是，所有需要的计算都是在MATHEMATICA \（^ {TM} \）中完成的。应用四个示例性例子研究该方法的准确性和适用性。