Numerical Solution of Boundary Value Problems by Wavelet-Based Galerkin Method

Abstract

Differential equations are the formulation of scientific theory for many real-world physical problems. Boundary value problems (BVPs) occur frequently in the fields of engineering and science, such as gas dynamics, nuclear physics, atomic structures, and chemical reactions.  In most cases, BVPs do not always find the exact solutions to these problems. Boubaker wavelets are wavelet functions derived from Boubaker polynomials. They serve as an effective numerical tool for tackling a range of scientific and engineering problems, including differential and variational equations. Their strength lies in generating accurate approximate solutions by transforming complicated equations into simpler linear systems. In this paper, a wavelet-based Galerkin method using Boubaker wavelets for the numerical solution of BVPs is proposed. Here, Boubaker wavelets are used as weight functions that are the assumed basis elements that allow us to obtain the numerical solution of the BVPs. The numerical results from the proposed method are compared with the exact solution to assess accuracy against existing schemes (Galerkin method using other wavelets, such as Laguerre and Fibonacci wavelets). Some BVPs are taken to demonstrate the validity and applicability of the proposed method.