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DOI：http://dx.doi.org/10.26855/jamc.2022.12.003

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

Date: November 3,2022 |Hits: 3204 Download PDF How to cite this paper

Shovan Sourav Datta Pranta1, Hazrat Ali1, Md. Shafiqul Islam1,*, Md. Shariful Islam2

1Department of Applied Mathematics, University of Dhaka, Dhaka-1000, Bangladesh.

2Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh.

*Corresponding author: Md. Shafiqul Islam

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.

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How to cite this paper

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

How to cite this paper:  Shovan Sourav Datta Pranta, Hazrat Ali, Md. Shafiqul Islam, Md. Shariful Islam. (2022) On the Numerical Treatment of 2D Nonlinear Parabolic PDEs by the Galerkin Method with Bivariate Bernstein Polynomial Bases. Journal of Applied Mathematics and Computation6(4), 410-422.

DOI: http://dx.doi.org/10.26855/jamc.2022.12.003