Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term

Essam R. El-Zahar

doi:10.1155/2016/1015634

Abstract

A reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and Differential Transform Method (DTM). First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order singular perturbation problems with a discontinuous source term.

Highlights

Many mathematical problems that model real-life phenomena cannot be solved completely by analytical means
The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order singular perturbation problems with a discontinuous source term
The results show that the method is a reliable and convenient asymptotic semianalytical method for treating high-order singular perturbation problems with a discontinuous source term

Summary

Introduction

Many mathematical problems that model real-life phenomena cannot be solved completely by analytical means. Some authors have developed numerical methods for problems with discontinuous data which gives rise to an interior layer in the exact solution of the problem, in addition to the boundary layer at the outflow boundary point. Most notable among these methods are piecewise-uniform mesh finite difference method [11,12,13,14] and fitted mesh finite element method [15, 16] for third- and fourth-order SPBVPs with a discontinuous source term. A reliable algorithm is presented to develop piecewise approximate analytical solutions of thirdand fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The results show that the method is a reliable and convenient asymptotic semianalytical method for treating high-order singular perturbation problems with a discontinuous source term

Differential Transform Method for ODE System

Description of the Method

Illustrating Examples

Conclusion