Numerical Solution of One-Dimensional Differential Equations by Weighted Residual Method through Taylor Wavelets

Abstract

Engineering problems are typically described using specific equations and conditions that define the boundaries of the problem. The numerical approach facilitates the resolution of complex issues through elementary operations. In contrast to analytical methods, a significant benefit of numerical methods is their ease of implementation on contemporary computers, allowing for rapid solutions compared to those obtained through analytical techniques. Galerkin's method belongs to a broader category of numerical techniques called weighted residual methods.  Also, wavelets emerged independently across various disciplines, including mathematics, quantum physics, electrical engineering, and seismic geology. This paper presents the weighted residual method, focusing on Galerkin's approach, for solving one-dimensional differential equations using Taylor wavelets. In this approach, the weight functions employed are Taylor wavelets, which serve as building blocks to facilitate the calculation of numerical solutions for one-dimensional differential equations. Illustrative examples are included to show the efficiency and precision of the technique. The numerical results show that the proposed method is more reasonable than the exact solution.