Numerical solution of a singularly perturbed Volterra integro-differential equation

We consider the singularly perturbed initial value problem for a linear first order Volterra integro-differential equation with delay. Our purpose is to construct and analyse a numerical method with uniform convergence in the perturbation parameter. The numerical solution of this problem is discretised using an implicit difference rules for differential part and the composite numerical quadrature rules for integral part. On a layer-adapted mesh error estimations for the approximate solution are established. Numerical examples supporting the theory are presented.

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“…Various approximating aspects for singularly perturbed VIDE's have also been investigated in [2,5,6,18,20,26,28,31].…”

Section: Introductionmentioning

confidence: 99%

A finite-difference method for a singularly perturbed delay integro-differential equation

Kudu¹,

Amirali²,

Amiraliyev³

2016

Journal of Computational and Applied Mathematics

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“…For a survey of early results in the theoretical analysis of singularly perturbed Volterra integro-differential equations (VIDEs) and in the numerical analysis and implementation of various techniques for these problems we refer to the book [11]. An analysis of approximate methods when applied to singularly perturbed VIDEs can also be found in [12,18,20].…”

Section: Introductionmentioning

confidence: 99%

On the Volterra Delay-Integro-Differential equation with layer behavior and its numerical solution

Amiraliyev¹,

Yapman²

2019

MMN

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In this paper, we analyze the convergence of the fitted mesh method applied to singularly perturbed Volterra delay-integro-differential equation. Our mesh comprises a special nonuniform mesh on the first subinterval and uniform mesh on another part. Error estimates are obtained using difference analogue of Gronwall's inequality with delay. A numerical test that confirms the theoretical results is presented.

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“…A general overview of several techniques to integrate Volterra/Fredholm integral or integro-differential equations can be found in [1,11,12,14,17]. In 2006 Bijura [5] demonstrated the existence of the initial layers whose thickness is not of order of magnitude O(ε), ε −→ 0, and developed approximate solutions using the initial layer theory In [16], Ş evgin studied the convergence properties of a difference scheme for singularly perturbed Volterra integro-differential equations on a graded mesh. Zhongdi and Lifeng [18] used the midpoint difference operator along with trapezoidal integration on a piecewise uniform Shishkin mesh to develop the numerical method for (1.1)-(1.2).…”

Section: Introductionmentioning

confidence: 99%

“…Thus, the approach to construct difference problems and analyze the error for approximate the solutions is analogous to the ones from [2], [10] and [16] and based upon some quadrature rules introduced by Amiraliyev [3]. An extension and summary of these rules are given in Amiraliyev and Mamedou [4].…”

Section: Introductionmentioning

confidence: 99%

New Parameter-Uniform Discretisations of Singularly Perturbed Volterra Integro-Differential Equations

Iragi¹,

Munyakazi²

2018

Appl. Math. Inf. Sci

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We design and analyse two numerical methods namely a fitted mesh and a fitted operator finite difference methods for solving singularly perturbed Volterra integro-differential equations. The fitted mesh method we propose is constructed using a finite difference operator to approximate the derivative part and some suitably chosen quadrature rules for the integral part. To obtain a parameter-uniform convergence, we use a piecewise-uniform mesh of Shishkin type. On the other hand, to construct the fitted operator method, the Volterra integro-differential equation is discretised by introducing a fitting factor via the method of integral identity with the use of exponential basis function along with interpolating quadrature rules [2]. The two methods are analysed for convergence and stability. We show that the two methods are robust with respect to the perturbation parameter. Two numerical examples are solved to show the applicability of the proposed schemes.

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Numerical solution of a singularly perturbed Volterra integro-differential equation

Cited by 24 publications

References 24 publications

A finite-difference method for a singularly perturbed delay integro-differential equation

A finite-difference method for a singularly perturbed delay integro-differential equation

On the Volterra Delay-Integro-Differential equation with layer behavior and its numerical solution

New Parameter-Uniform Discretisations of Singularly Perturbed Volterra Integro-Differential Equations

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