A numerical method on Bakhvalov-Shishkin mesh for Volterra integro-differential equations with a boundary layer

Hayriye Guckir Cakir ,Firat Cakir ,Musa Çakır

doi:10.31801/cfsuasmas.946910

Abstract

We construct a finite difference scheme for a first-order linear singularly perturbed Volterra integro-differential equation(SPVIDE) on Bakhvalov-Shishkin mesh. For the discretization of the problem, we use the integral identities and deal with the emerging integrals terms with interpolating quadrature rules which also yields remaining terms. The stability bound and the error estimates of the approximate solution are established. Further, we demonstrate that the scheme on Bakhvalov-Shishkin mesh is N ( O − 1 ) N(O−1) u n iformly convergent, where N is the mesh parameter. The numerical results are also provided for a couple of examples.

Highlights

In this present work, we are consider the following class of the singularly perturbed linear Volterra integro-differential equations (SPVIDEs) xLu := εu′ + a(x)u + λ K(x, t)u(t)dt = f (x), x ∈ I = [0, l], (1)subject to u(0) = A, (2)where 0 < ε ≪ 1 is a small perturbation parameter
A finite difference scheme is utilized to examine the numerical solutions of a non-linear VIDE in [11]
Before we proceed to the definition of the mesh points and discretization of the problem we provide the notation we use throughout the paper

Summary

Introduction

We are consider the following class of the singularly perturbed linear Volterra integro-differential equations (SPVIDEs) x. In [5], a specific integro-differential equation with a boundary layer which describes filament stretching process is considered and the leading order behavior of the problem is examined by an asymptotic method. A finite difference scheme on Bakhvalov-Shishkin mesh is utilized to deal with a singularly perturbed boundary value problem in [10].

Asymptotic Behavior of the Solution

Difference Scheme

C K 2 hj

Algorithm and Numerical Results

Conclusion