A B-Polynomial Approach to Approximate Solutions of PDEs with Multiple Initial Conditions - Muhammad I. Bhatti; Md. Habibur Rahman (Axioms)
In this article, we present a novel B-Polynomial Approach for approximating solutions to partial differential equations (PDEs), addressing the multiple initial conditions. Our method stands out by utilizing two-dimensional Bernstein polynomials (B-polynomials) in conjunction with their operational matrices to effectively manage the complexity associated with PDEs. This approach not only enhances the accuracy of solutions but also provides a structured framework for tackling various boundary conditions. The PDE is transformed into a system of algebraic equations, which are then solved to approximate the PDE solution. The process is divided into two main steps: First, the PDE is integrated to incorporate all initial and boundary conditions. Second, we express the approximate solution using B-polynomials and determine the unknown expansion coefficients via the Galerkin finite element method. The accuracy of the solution is assessed by adjusting the number of B-polynomials used in the expansion. The absolute error is estimated by comparing the exact and semi-numerical solutions. We apply this method to several examples, presenting results in tables and visualizing them with graphs. The approach demonstrates improved accuracy as the number of B-polynomials increases, with CPU time increasing linearly. Additionally, we compare our results with other methods, highlighting that our approach is both simple and effective for solving multidimensional PDEs imposed with multiple initial and boundary conditions.