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

Abstract

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.

A second order numerical method for singularly perturbed Volterra integro-differential equations with delay

“…The construction of the difference scheme is a combination of hybrid difference scheme on a Shishkin-type mesh and appropriate quadrature rules. Error estimates are obtained by using the truncation error estimate techniques and a discrete analogue of Grönwall's inequality and it is shown that the scheme is O(N −2 ln 2 N) order convergent, which improves the numerical results given in [7,[12][13][14]. Numerical experiments are presented to support the theoretical result.…”

“…Kudu et al [7] proposed an implicit finite difference method on a piecewise uniform Shishkin-type mesh with appropriate quadrature rules for a singularly perturbed delay integro-differential equation and showed that the scheme is almost first-order convergent. Amiraliyev and Yapman [12] used a fitted mesh method to solve a singularly perturbed Volterra delay-integro-differential equation and indicated that the scheme is also almost first-order convergent. Yapman et al [13] developed a numerical method of integral identities with the use of exponential basis functions and interpolating quadrature rules for a singularly perturbed nonlinear Volterra integro-differential equation with delay and proved that the scheme is first-order uniform convergent.…”

An efficient numerical method for a singularly perturbed Volterra-Fredholm integro-differential equation

“…Proof. The proof is done by similar approach as in [5,12]. Now, we turn to establishment of the difference scheme.…”

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

“…On account of 2) , it is found that [3,6,19,29,33]. b) Now, we consider the node points of Bakhvalov type mesh.…”

“…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.…”