Quantitative Analysis of Error Propagation and Computational Efficiency in Numerical Solutions of Boundary Value Problems

Abstract

This study presents a comparative analysis of three widely used numerical methods, the Shooting method, the Galerkin method, and the Modified Galerkin method for solving boundary value problems (BVPs). The Shooting method is applied by converting the BVP into an initial value problem, solved by a Runge-Kutta scheme, and the Galerkin and Modified Galerkin methods are developed in a weighted residual form. The methods are used on a variety of problems, such as application-driven models like population dynamics and epidemiological systems. The accuracy of each method is assessed by an error analysis based on the root mean square (RMS) error. Besides accuracy, the computation time is also a performance metric used to analyze the trade-off between efficiency and precision. The numerical data indicate that the Shooting method is more likely to give a better accuracy in complex problems, and the Modified Galerkin method is more favorable to balance accuracy and computational cost. In contrast, the standard Galerkin method exhibits comparatively lower accuracy under the considered problem settings. This paper points out the comparative weaknesses and strengths of these techniques and can help in understanding how these techniques can be used to solve BVPs.

Jagannath University Journal of Science, Volume 12, Number 1, Jun. 2025, pp. 159−176