The purpose of the research is to find the numerical solutions to the system of time dependent nonlinear parabolic partial differential equations (PDEs) utilizing the Modified Galerkin Weighted Residual Method (MGWRM) with the help of modified Bernstein polynomials. An approximate solution of the system has been assumed in accordance with the modified Bernstein polynomials. Thereafter, the modified Galerkin method has been applied to the system of nonlinear parabolic PDEs and has transformed the model into a time dependent ordinary differential equations system. Then the system has been converted into the recurrence equations by employing backward difference approximation. However, the iterative calculation is performed by using the Picard Iterative method. A few renowned problems are then solved to test the applicability and efficiency of our proposed scheme. The numerical solutions at different time levels are then displayed numerically in tabular form and graphically by figures. The comparative study is presented along with L2 norm, and L infinity norm.

翻译：本研究旨在利用修正的Galerkin加权残量法（MGWRM），结合修正的Bernstein多项式，求解含时非线性抛物型偏微分方程组（PDEs）的数值解。首先，根据修正的Bernstein多项式假设得到方程组的近似解。随后，将修正的Galerkin方法应用于非线性抛物型PDEs系统，将模型转化为含时常微分方程组。接着，通过采用向后差分近似，将此系统转化为递推方程。进而，利用Picard迭代法完成迭代计算。最后，求解若干经典问题以检验所提格式的适用性和有效性。不同时间层上的数值解以表格形式和图形方式展示。对比研究基于L2范数和L无穷范数进行。