On the Numerical Treatment of 2D Nonlinear Parabolic PDEs by the Galerkin Method with Bivariate Bernstein Polynomial Bases

Abstract

In this study, numerical solutions are obtained for the time-dependent two-dimensional nonlinear parabolic partial differential equations (PDEs) with initial and Dirichlet boundary conditions. In assessing spatial derivatives, we employ the modified Galerkin method with the aid of Green's theorem, which minimizes the derivatives' order and incorporates boundary conditions. In the trial function, we use bivariate Bernstein polynomial bases. All the initial and boundary conditions are handled carefully by suitable transformation. Further, we exploit an iterative α-family approximation, especially the Crank Nicolson scheme, to take care the time derivative. Applying the proposed technique to a variety of nonlinear 2D parabolic PDEs, such as the 2D Burger's equation and the 2D Convection-Diffusion Reaction equation, the numerical results are presented in the form of tables and figures. The numerical results provide conclusive evidence that the technique being proposed is accurate and effective.