A Numerical Method for Nonlinear Singularly Perturbed Multi-Point Boundary Value Problem

Musa Çakır,Derya Arslan

doi:10.4236/jamp.2016.46119

Abstract

We consider a uniform finite difference method for nonlinear singularly perturbed multi-point boundary value problem on Shishkin mesh. The problem is discretized using integral identities, interpolating quadrature rules, exponential basis functions and remainder terms in integral form. We show that this method is the first order convergent in the discrete maximum norm for original problem (independent of the perturbation parameter e). To illustrate the theoretical results, we solve test problem and we also give the error distributions in the solution in Table 1 and Figures 1-3.

Highlights

In this paper we shall consider singularly perturbed multi-point nonlinear problem−ε 2u′′ + f ( x,u=) 0, 0 ≤ x ≤ 1, (1) u(0) = 0 (2) k0u m = ∑ kiu + km+1
The results show that the uniform finite difference method on Shishkin mesh is more powerful method than other methods for nonlinear singularly perturbed multi-point boundary value problem
We present some properties of the mesh function g ( x) defined on ωN, which is needed for analysis of the numerical solution

Summary

Introduction

In this paper we shall consider singularly perturbed multi-point nonlinear problem. How to cite this paper: Çakır, M. and Arslan, D. (2016) A Numerical Method for Nonlinear Singularly Perturbed Multi-Point Boundary Value Problem. (2016) A Numerical Method for Nonlinear Singularly Perturbed Multi-Point Boundary Value Problem. Amiraliyev and Çakır [4] applied the difference method on a Shishkin mesh to the singularly perturbed three-point boundary value problem. Amiraliyev and Çakır [6] studied numerical solution of the singularly perturbed problem with nonlocal boundary condition. Amiraliyeva, Erdoğan and Amiraliyev [9] applied a uniform numerical method for dealing with a singularly perturbed delay initial value problem. Çakır [14] studied uniform second-order difference method for a singularly perturbed three-point boundary value problem. Geng [15] applied a numerical algorithm for nonlinear multi-point boundary value problems. The results show that the uniform finite difference method on Shishkin mesh is more powerful method than other methods for nonlinear singularly perturbed multi-point boundary value problem

The Continuous Problem

Discretizaton and Non-Uniform Mesh

Error Analysis

Numerical Example

Conclusion