A stable numerical method for convection dominated nonlinear Volterra integro-differential equations

Yidnekachew Abebe Wondmu; Muath Awadalla; Meraa Arab; Mesfin Mekuria Woldaregay; Yidnekachew Abebe Wondmu; Muath Awadalla; Meraa Arab; Mesfin Mekuria Woldaregay

doi:10.3934/math.2026699

AIMS Mathematics

2026, Volume 11, Issue 6: 17062-17092. doi: 10.3934/math.2026699

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Research article

A stable numerical method for convection dominated nonlinear Volterra integro-differential equations

1.
Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia

Received: 31 January 2026 Revised: 21 May 2026 Accepted: 03 June 2026 Published: 12 June 2026
MSC : Primary 65L11, 65L12, 45J05, 65L20, 65R20, 34E15

This paper presented a stable numerical method for solving convection dominated singularly perturbed nonlinear Volterra integro-differential equations. The equation involved a small perturbation parameter $ \varepsilon \in (0, 1] $, which leads to a boundary layer in the solution, causing sharp gradients that classical numerical methods struggle to resolve without excessive computational effort. To handle this challenge, an exponentially fitted difference method was proposed for the differential part, which incorporated $ \varepsilon $ into the discrete operator to accurately capture the boundary layer on coarse meshes. The Volterra integral term was approximated using the composite Trapezoidal rule. The stability and convergence analysis confirmed that the proposed method was uniformly convergent with respect to $ \varepsilon $, preserving accuracy even for small values of $ \varepsilon $. Numerical experiments were implemented on four test problems to validate the theoretical results, demonstrating the accuracy, uniform convergence, and the method's ability to handle a boundary layer. We extended the method to handle a nonlinear problem by incorporating Newton's linearization technique. The computed result showed that the proposed method was accurate and stable under strong boundary layer condition.
- Volterra integro-differential equation,
- exponentially fitted method,
- nonlinear,
- trapezoidal method,
- stability,
- boundary layer
Citation: Yidnekachew Abebe Wondmu, Muath Awadalla, Meraa Arab, Mesfin Mekuria Woldaregay. A stable numerical method for convection dominated nonlinear Volterra integro-differential equations[J]. AIMS Mathematics, 2026, 11(6): 17062-17092. doi: 10.3934/math.2026699

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Abstract

This paper presented a stable numerical method for solving convection dominated singularly perturbed nonlinear Volterra integro-differential equations. The equation involved a small perturbation parameter $ \varepsilon \in (0, 1] $, which leads to a boundary layer in the solution, causing sharp gradients that classical numerical methods struggle to resolve without excessive computational effort. To handle this challenge, an exponentially fitted difference method was proposed for the differential part, which incorporated $ \varepsilon $ into the discrete operator to accurately capture the boundary layer on coarse meshes. The Volterra integral term was approximated using the composite Trapezoidal rule. The stability and convergence analysis confirmed that the proposed method was uniformly convergent with respect to $ \varepsilon $, preserving accuracy even for small values of $ \varepsilon $. Numerical experiments were implemented on four test problems to validate the theoretical results, demonstrating the accuracy, uniform convergence, and the method's ability to handle a boundary layer. We extended the method to handle a nonlinear problem by incorporating Newton's linearization technique. The computed result showed that the proposed method was accurate and stable under strong boundary layer condition.

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