Galerkin-Bernstein Approximations of the System of Time Dependent Nonlinear Parabolic PDEs

TL;DR

This paper develops a numerical method using Modified Galerkin and Bernstein polynomials to solve nonlinear parabolic PDE systems, transforming them into ODEs and employing iterative schemes for solutions.

Contribution

It introduces a novel approach combining Modified Galerkin and Bernstein polynomials for efficiently solving nonlinear parabolic PDEs.

Findings

Numerical solutions are obtained and validated at different time levels.

The method shows good accuracy through L2 and L infinity norm comparisons.

The approach effectively transforms PDEs into solvable ODE systems.

Abstract

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.