A uniformly convergent numerical method for a singularly perturbed Volterra integro-differential equation

Iragi, Bakulikira C.; Munyakazi, Justin B.

doi:10.1080/00207160.2019.1585828

Cited by 28 publications

(4 citation statements)

References 18 publications

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“…The uniqueness of the solution is guaranteed by this discrete maximum principle. The existence follows easily since, as for linear problems, the existence of the solution is implied by its uniqueness [ 12 ]. The discrete maximum principle enables us to prove the next lemma which provides the boundedness of the solution.…”

Section: Stability and Convergence Analysismentioning

confidence: 99%

“…Due to this, the numerical treatment of singularly perturbed problems presents severe difficulties that have to be addressed to ensure accurate numerical solutions [ 4 ]. As a result, in recent years, few works on the numerical solution of singularly perturbed Fredholm/Volterra integral equations have been recorded in the literature [ 11 , 12 ]. Durmaz et al [ 8 ] developed a fitted difference scheme on Shishkin mesh using interpolating quadrature rules and an exponential basis function for the numerical treatment of the singularly perturbed Fredholm integro-differential equation with mixed boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Solving singularly perturbed fredholm integro-differential equation using exact finite difference method

Badeye,

Woldaregay,

Dinka

2023

BMC Res Notes

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Objectives In this paper, a numerical scheme is designed for solving singularly perturbed Fredholm integro-differential equation. The scheme is constructed via the exact (non-standard) finite difference method to approximate the differential part and the composite Simpson’s 1/3 rule for the integral part of the equation. Result The stability and uniform convergence analysis are demonstrated using solution bound and the truncation error bound. For three model examples, the maximum absolute error and the rate of convergence for different values of the perturbation parameter and mesh size are tabulated. The computational result shows, the proposed method is second-order uniformly convergent which is in a right agreement with the theoretical result.

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Section: Stability and Convergence Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Solving singularly perturbed fredholm integro-differential equation using exact finite difference method

Badeye,

Woldaregay,

Dinka

2023

BMC Res Notes

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“…Some existence and uniqueness results about singularly perturbed problems have been given in [15,23]. In recent times, notable techniques and various numerical schemes have been presented for singularly perturbed integro-differential equations (see [2,6,9,10,13,17,19,25,29,[33][34][35]). Our aim in this paper is to present a uniform numerical method for solving singularly perturbed nonlinear integro-differential equations and compare the obtained results on Bakhvalov and Shishkin type meshes.…”

Section: Introductionmentioning

confidence: 99%

A numerical comparative study for the singularly perturbed nonlinear Volterra-Fredholm integro-differential equations on layer-adapted meshes

Gunes,

Cakir

2024

MMN

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This article deals with the singularly perturbed nonlinear Volterra-Fredholm integro-differential equations. Firstly, some priori bounds are presented. Then, the finite difference scheme is constructed on non-uniform mesh by using interpolating quadrature rules [5] and composite numerical integration formulas. The error estimates are derived in the discrete maximum norm. Finally, theoretical results are performed on two examples and they are compared for both Bakhvalov (B-type) and Shishkin (S-type) meshes.

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“…Exponentially fitted difference scheme has been constructed on a Shishkin mesh for second order SPFIDEs in [11]. The stability and convergence of the difference scheme have been analyzed for SPVIDEs in [22]. Boundary value problems of SPFIDEs have been investigated in [9].…”

Section: Introductionmentioning

confidence: 99%

A new difference method for the singularly perturbed Volterra-Fredholm integro-differential equations on a Shishkin mesh

Cakir

Güneş

2022

Hacettepe Journal of Mathematics and Statistics

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In this research, the finite difference method is used to solve the initial value problem of linear first order Volterra-Fredholm integro-differential equations with singularity. By using implicit difference rules and composite numerical quadrature rules, the difference scheme is established on a Shishkin mesh. The stability and convergence of the proposed scheme are analyzed and two examples are solved to display the advantages of the presented technique.

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A uniformly convergent numerical method for a singularly perturbed Volterra integro-differential equation

Cited by 28 publications

References 18 publications

Solving singularly perturbed fredholm integro-differential equation using exact finite difference method

Solving singularly perturbed fredholm integro-differential equation using exact finite difference method

A numerical comparative study for the singularly perturbed nonlinear Volterra-Fredholm integro-differential equations on layer-adapted meshes

A new difference method for the singularly perturbed Volterra-Fredholm integro-differential equations on a Shishkin mesh

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