Highly accurate multi-Galerkin method for systems of nonlinear Volterra integral equations based on Kumar–Sloan approach

Krishna Murari Malav, Kapil Kant, Samiran Chakraborty and Joydip Dhar

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 248, issue C, 686-712

Abstract: In this article, we apply the multi-Galerkin method based on the Kumar–Sloan approach to solve the system of Volterra–Hammerstein integral equations with smooth and weakly singular kernels using piecewise polynomial basis functions. First, we develop the mathematical formulation of the multi-Galerkin method using piecewise polynomial approximation subspace to solve such systems. Then we derive error estimates and establish improved convergence rates. To do this, we construct the uniform mesh for smooth kernels and the graded mesh for weakly singular kernels on [0,1]. Moreover, we show that the Kumar–Sloan-based multi-Galerkin method obtains superconvergence results without using its iterated method. Hence, it is noticed that the multi-Galerkin method demonstrates improved performance over the Galerkin method, while maintaining the same system size for both methods. Finally, numerical results are provided to validate the theoretical findings and confirm the expected convergence rate.

Keywords: System of Volterra–Hammerstein integral equations; Kumar–Sloan approach; Multi-Galerkin method; Piecewise polynomials; Smooth kernels; Weakly singular kernels; Superconvergence results (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:248:y:2026:i:c:p:686-712

DOI: 10.1016/j.matcom.2026.05.001

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