An efficient numerical method for a singularly perturbed Fredholm integro-differential equation with integral boundary condition

Muhammet Enes Durmaz; Ilhame Amirali; Gabil M Amiraliyev

doi:10.1007/s12190-022-01757-4

An efficient numerical method for a singularly perturbed Fredholm integro-differential equation with integral boundary condition

J Appl Math Comput. 2023;69(1):505-528. doi: 10.1007/s12190-022-01757-4. Epub 2022 Jun 9.

Authors

Muhammet Enes Durmaz¹, Ilhame Amirali^#², Gabil M Amiraliyev^#³

Affiliations

¹ Department of Information Technology, Kırklareli University, Kırklareli, 39100 Turkey.
² Department of Mathematics, Faculty of Arts and Sciences, Düzce University, Düzce, 81620 Turkey.
³ Department of Mathematics, Faculty of Arts and Sciences, Erzincan Binali Yıldırım University, Erzincan, 24100 Turkey.

^# Contributed equally.

Abstract

In this paper, a linear singularly perturbed Fredholm integro-differential initial value problem with integral condition is being considered. On a Shishkin-type mesh, a fitted finite difference approach is applied using a composite trapezoidal rule in both; in the integral part of equation and in the initial condition. The proposed technique acquires a uniform second-order convergence in respect to perturbation parameter. Further provided the numerical results to support the theoretical estimates.

Keywords: Finite difference scheme; Fredholm integro-differential equation; Integral boundary condition; Shishkin mesh; Singular perturbation; Uniform convergence.