A comparative study of analytical and numerical methods for solving systems of nonlinear Volterra integral equations with applications Comparative study of methods for nonlinear Volterra equations

Abstract

This paper presents a detailed comparison of three relatively recent methods for the numerical solution of systems of Nonlinear Volterra Integral Equations of the second kind (NVIEs-II): the Modified Adomian Decomposition Method (MADM), the Hussein-Jassim Method (H-JM), and the Cubic Non-Polynomial Spline Function Method (CNPSFM). The objective of this study is to evaluate the performance of these methods in terms of accuracy, convergence, and numerical stability. To achieve this, all three methods are applied to standard benchmark problems with known exact solutions, enabling quantitative assessment. The numerical results reveal distinct performance characteristics for each method. Both MADM and H-JM demonstrate excellent performance, yielding solutions with high accuracy and very low errors, occasionally approaching machine precision. MADM exhibits rapid convergence, while H-JM provides robust numerical stability and ease of implementation. CNPSFM displays good numerical stability and accurately captures the overall solution behavior; however, it produces relatively larger errors, particularly as the integration interval lengthens. This comparison concludes that the optimal choice among these methods is highly problem-specific. MADM and H-JM are best suited for high-precision applications requiring analytical insight (e.g., quantum mechanics or population dynamics), whereas CNPSFM remains viable for applications prioritizing solution smoothness over absolute accuracy. This study provides practical, evidence-based recommendations to assist researchers and engineers in selecting appropriate solvers for real-world systems modeled by NVIEs-II. Future research should extend these methods to systems with singularities and/or delays, which have been underexplored in the current literature. Another promising direction involves developing hybrid approaches that integrate artificial neural networks with traditional computational solvers.