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A robust numerical technique for solving non-linear Volterra integro-differential equations with boundary layer

Firat Cakir|Musa Cakir|Hayriye Guckir Cakir
대한수학회 (The Korean Mathematical Society)|2022.07|대한수학회논문집 (Communications of the KMS)|제37권 제3호|pp. 939-955|DOI 10.4134/CKMS.c210261

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참고문헌

키워드

uniform convergenceerror estimatesSingularly perturbedVIDEdifference schemesBakhvalov-Shishkin mesh
주제분류수학

초록

In this paper, we study a first-order non-linear singularly perturbed Volterra integro-differential equation (SPVIDE). We discretize the problem by a uniform difference scheme on a Bakhvalov-Shishkin mesh. The scheme is constructed by the method of integral identities with exponential basis functions and integral terms are handled with interpolating quadrature rules with remainder terms. An effective quasi-linearization technique is employed for the algorithm. We establish the error estimates and demonstrate that the scheme on Bakhvalov-Shishkin mesh is $O(N^{-1})$ uniformly convergent, where $N$ is the mesh parameter. The numerical results on a couple of examples are also provided to confirm the theoretical analysis.

Abstract

In this paper, we study a first-order non-linear singularly perturbed Volterra integro-differential equation (SPVIDE). We discretize the problem by a uniform difference scheme on a Bakhvalov-Shishkin mesh. The scheme is constructed by the method of integral identities with exponential basis functions and integral terms are handled with interpolating quadrature rules with remainder terms. An effective quasi-linearization technique is employed for the algorithm. We establish the error estimates and demonstrate that the scheme on Bakhvalov-Shishkin mesh is $O(N^{-1})$ uniformly convergent, where $N$ is the mesh parameter. The numerical results on a couple of examples are also provided to confirm the theoretical analysis.

참고문헌 (34)

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