Galerkin-Bernstein Approximations for the System of Third-Order Nonlinear Boundary Value Problems

TL;DR

This paper introduces a numerical method using Bernstein polynomial-based Galerkin approach to solve third-order nonlinear boundary value problems, demonstrating accuracy and convergence through examples and comparisons.

Contribution

The paper develops a novel Bernstein polynomial-based Galerkin method for third-order nonlinear BVPs, including matrix formulation and reduction techniques for complex problems.

Findings

Method shows good accuracy on examples

Solutions converge monotonically to exact solutions

Performance is satisfactory compared to existing methods

Abstract

This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We derive mathematical formulations in matrix form, in details, by exploiting Bernstein polynomials as basis functions. A reasonable accuracy is found when the proposed method is used on few examples. At the end of the study, a comparison is made between the approximate and exact solutions, and also with the solutions of the existing methods. Our results converge monotonically to the exact solutions. In addition, we show that the the derived formulations may be applicable by reducing higher order complicated BVP into a lower order system of BVPs, and the performance of the numerical solutions is satisfactory.