Accelerated Exponentially Fitted Operator Method for Singularly Perturbed Problems with Integral Boundary Condition

Habtamu Garoma Debela,Gemechis File Duressa

doi:10.1155/2020/9268181

Habtamu Garoma Debela, Gemechis File Duressa

Open Access

https://doi.org/10.1155/2020/9268181

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Abstract

In this paper, we consider a class of singularly perturbed differential equations of convection diffusion type with integral boundary condition. An accelerated uniformly convergent numerical method is constructed via exponentially fitted operator method using Richardson extrapolation techniques and numerical integration methods to solve the problem. The integral boundary condition is treated using numerical integration techniques. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be ε-uniformly convergent.

Highlights

Boundary value problems involving integral boundary conditions have received considerable attention in recent years [1, 2]
Discussion and Conclusion is study introduces the uniformly convergent numerical method based on the exponential fitted operator method for solving singularly perturbed boundary value problems with integral boundary conditions. e behavior of the continuous solution of the problem is studied and the derivatives of the solution are bounded. e numerical scheme is developed on a uniform mesh. e integral boundary condition is treated using numerical integration techniques, namely, Simpson’s rule; the results are compared . e stability of the developed scheme is established and its uniform convergence is proved
To validate the applicability of the method, a model problem/example is considered for numerical experimentation for different values of the perturbation parameter and mesh points. e numerical results are tabulated in terms of maximum absolute errors, numerical rate of convergence, and uniform errors and compared with the results of the previously developed numerical methods existing in the literature (Table 2)

Summary

Introduction

Boundary value problems involving integral boundary conditions have received considerable attention in recent years [1, 2]. The methods or algorithms developed so far mainly concerned with the regular cases (i.e., when the boundary layers are absent). E solutions of the problems with boundary layer undergo rapid changes within very thin layers near the boundary or inside the problem domain [2, 8,9,10], and classical numerical methods for solving such problems are unstable and fail to give good results when the perturbation parameter is small (i.e., for h ≥ ε) [10]. As far as the researchers’ knowledge is concerned numerical solution of the singularly perturbed boundary value problem containing integral boundary condition via the accelerated exponential fitted operator method is first being considered. This paper proposed a uniformly convergent numerical method based on exponential fitted operator and numerical integration methods to solve the problem under consideration

Statement of the Problem

Formulation of the Method

Uniform Convergence Analysis

Numerical Example and Results