Singularly Perturbed Nonlinear Volterra Integro-Differential Equations with Fading Memory: Analysis and Numerical Approximation

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DOI: https://doi.org/10.28924/2291-8639-24-2026-337
Published: Oct 15, 2026
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Phaisatcha Inpoonjai, Singularly Perturbed Nonlinear Volterra Integro-Differential Equations with Fading Memory: Analysis and Numerical Approximation, Int. J. Anal. Appl., 24 (2026), 337.

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Phaisatcha Inpoonjai

Abstract

This paper investigates a class of singularly perturbed nonlinear Volterra integro-differential equations with fading memory effects. The model involves nonlinear reaction terms together with Volterra-type integral operators, leading to multiscale behavior characterized by the presence of an initial boundary layer. Under suitable assumptions, the existence, uniqueness, and positivity of solutions are established. An asymptotic analysis is carried out through the associated reduced problem and boundary layer correction, revealing the two-scale structure of the solution. Motivated by this analysis, a layer-adapted numerical method based on a Shishkin-type mesh is constructed. The differential term is discretized using a backward difference approximation, while the Volterra integral term is approximated by a composite trapezoidal rule. The resulting nonlinear discrete system is solved by Newton's iterative method. The proposed scheme is shown to be consistent and stable, and a convergence estimate is established in the asymptotic regime corresponding to sufficiently fine mesh discretizations. Numerical experiments confirm the theoretical results and demonstrate the effectiveness of the method in accurately resolving the boundary layer behavior.

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