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

Abstract

The scope of this study is to establish an effective approximation method for linear first order singularly perturbed Volterra-Fredholm integro-differential equations. The finite difference scheme is constructed on Shishkin mesh by using appropriate interpolating quadrature rules and exponential basis function. The recommended method is second order convergent in the discrete maximum norm. Numerical results illustrating the preciseness and computationally attractiveness of the proposed method are presented.

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

“…, which points to the proof of (2.4). Now, we show the proof of the relation (2.5) (see [9,11,13,29]). From (2.6), we write…”

“…In [31], different variational techniques have been proposed for Volterra-Fredholm integro-differential equations. Furthermore, the linear form of the problem (1.1)-(1.2) has been considered by using finite difference schemes in [9,13].…”

“…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.…”

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

“…Similar to the approaches for an integral part of the problem ( 14) the initial condition of (1) is discretized as, In a similar manner, we obtain the errors in the remaining sub-interval becomes, R 3 = −1 180 h 4 c 4 (ξ )u 4 (ξ ) . Based on (15) and (16) we propose the following difference scheme for approximating the problem (1) ( 14)…”

“…For the purpose of getting accurate numerical results, the literature is replete with material on the numerical handling of these issues. Durmaz et al in [7][8][9][10][11][12][13][14][15] considered different forms of the singularly perturbed integro-differential equations. They used different approaches to approximate the differential parts of the equations and used the composite trapezoidal rule for the integral part of the equations.…”