Initial-layer theory and model equations of Volterra type

Bijura, Angelina M.

doi:10.1093/imamat/hxh113

Cited by 3 publications

(1 citation statement)

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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%

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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