A Fitted Approximate Method for a Volterra Delay-Integro-Differential Equation with Initial Layer

Abstract

This study is concerned with the finite-difference solution of singularly perturbed initial value problem for a linear first order Volterra integrodifferential equation with delay. The method is based on the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder terms in integral form. The emphasis is on the convergence of numerical method. It is shown that the method displays uniform convergence in respect to the perturbation parameter. Numerical results are also given.

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

“…Amiraliyev et al recently constructed an exponential-difference scheme with an accuracy of O N −1 for the first-order linear singularly perturbed Fredholm integro-differential equation (SPFIDE) on a uniform grid in [1], and finite difference scheme with an accuracy of O N −2 ln N on a Shishkin grid for the second-order linear SPFIDE in [12]. The first and the second order difference schemes were proposed in [4,34]. In recent years, many authors have applied different methods such as homotopy analysis method, modified variational iteration method, Adomian decomposition method that is named Laplace discrete Adomian decomposition method, modified homotopy perturbation method to obtain approximate analytical solutions for Volterra, Fredholm, Volterra-Fredholm, fuzzy Volterra-Fredholm integro-differential equations in [8,9,15,[17][18][19].…”

A fitted numerical method for a singularly perturbed Fredholm integro-differential equation with discontinuous source term

“…Amiraliyev et al [2] showed that first-order convergence on a uniform mesh was obtained globally for a first-order equation considering an exponential fitting approach. A linearly accurate approach for singularly perturbed Volterra delayintegro-differential equations (SPVDIDEs) was presented in [3].…”

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

“…In [33], for SPVIDEs, a fitted mesh finite difference technique with Richardson extrapolation has been applied on piecewise-uniform Shishkin mesh. SPVIDEs with delay arguments have been discretized in [5].…”