A Study on the Numerical Accuracy of Galerkin, Modified Galerkin, Shooting and Homotopy Perturbation Methods in Solving Boundary Value Problems

Abstract

This study analyzes the behavior of four popular numerical methods of solving boundary-value problems (BVPs):
the Standard Galerkin method, the Modified Galerkin method, the Shooting method, and the Homotopy Perturbation Method
(HPM). The BVP is commonly used in scientific and engineering practice in fluid and thermal transport, micro- and bio-fluidic
systems, fluid-structure interaction, aerodynamics, and electromagnetic modelling. The evaluation of each technique was based on
the comparison of its numerical results and the errors with the analytical result. The precision and consistency were reflected by
detailed tables and graphic representations that point out solution behaviour and error patterns. The best accuracy was derived with
the Shooting method, followed by the Modified Galerkin method. The modified Galerkin method was more adaptable and stable
as compared to the standard Galerkin scheme, which showed bigger variations in its error outcomes. HPM, however, was observed
to have some irregularities, especially at the mid-point part of the solution domain. Therefore, the general comparison of this paper
explains how each numerical technique reacts to boundary-value problems and how much they can be applied in real computational
settings.