A Second-Order Post-processing Technique for Singularly Perturbed Volterra Integro-differential Equations

Panda, A K; Mohapatra, Jugal; Amirali, Ilhame

doi:10.1007/s00009-021-01873-8

Cited by 26 publications

(5 citation statements)

References 28 publications

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“…In [22], the fitted mesh finite difference method is considered for solving singularly perturbed Fredholm integro-differential equation. The derivative part is approximated using the upwind scheme and the integral part was estimated by the iterative quadrature rule.…”

Section: Introductionmentioning

confidence: 99%

A fitted operator numerical method for singularly perturbed Fredholm integro-differential equation with integral initial condition

Oljira,

Woldaregay

2024

BMC Res Notes

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Objectives In this paper, a uniformly convergent numerical scheme is proposed for solving a singularly perturbed Fredholm integro-differential equation with an integral initial condition. The equation involves a left boundary layer which makes it difficult to solve it using the standard numerical methods. A fitted operator finite difference method is used to approximate the differential part of the equation and the composite Simpson $$\frac{1}{3}$$ 1 3 rule is used for the integral parts of the equation and the initial condition. Result The stability bound and error estimation of the approximated solution are performed, to show the uniform convergence of the scheme with order one in the maximum norm. Numerical test examples are provided to calculate the maximum absolute errors, thrgence, and the uniform error for a couple of examples to support theoretical analysis.

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Section: Introductionmentioning

confidence: 99%

A fitted operator numerical method for singularly perturbed Fredholm integro-differential equation with integral initial condition

Oljira,

Woldaregay

2024

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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“…Significant efforts have been made in the last 50 years to treat the Volterra/Fredholm integral equations numerically [18,19,20,21,22,23,24,25,26,27,28,29]. More specically, in [30,31], a difference scheme of the exponential type on a uniform mesh is considered.…”

Section: Introductionmentioning

confidence: 99%

First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition

Amirali

Durmaz

Acar

et al. 2023

J. Phys.: Conf. Ser.

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In this work, we consider first-order singularly perturbed quasilinear Fredholm integro-differential equation with integral boundary condition. Building a numerical strategy with uniform ε-parameter convergence is our goal. With the use of exponential basis functions, quadrature interpolation rules and the method of integral identities, a fitted difference scheme is constructed and examined. The weight and remainder term are both expressed in integral form. It is shown that the method exhibits uniform first-order convergence of the perturbation parameter. Error estimates for the approximation solution are established and a numerical example is given to validate the theoretical findings.

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A Second-Order Post-processing Technique for Singularly Perturbed Volterra Integro-differential Equations

Cited by 26 publications

References 28 publications

A fitted operator numerical method for singularly perturbed Fredholm integro-differential equation with integral initial condition

A fitted operator numerical method for singularly perturbed Fredholm integro-differential equation with integral initial condition

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

First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition

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