Successive Complementary Expansion Method for Solving Troesch’s Problem as a Singular Perturbation Problem

Abstract

A simple and efficient method that is called Successive Complementary Expansion Method (SCEM) is applied for approximation to an unstable two-point boundary value problem which is known as Troesch’s problem. In this approach, Troesch’s problem is considered as a singular perturbation problem. We convert the hyperbolic-type nonlinearity into a polynomial-type nonlinearity using an appropriate transformation, and then we use a basic zoom transformation for the boundary layer and finally obtain a nonlinear ordinary differential equation that contains SCEM complementary approximation. We see that SCEM gives highly accurate approximations to the solution of Troesch’s problem for various parameter values. Moreover, the results are compared with Adomian Decomposition Method (ADM) and Homotopy Perturbation Method (HPM) by using tables.

An asymptotic-numerical hybrid method for singularly perturbed system of two-point reaction-diffusion boundary-value problems

“…is a uniformly valid approximation (UVA) [4][5][6]. For only one singularly perturbed differential equation, the uniformly valid SCEM approximation is in the regular form given by…”

“…Control theory, fluid mechanics, quantum mechanics, combustion theory, signal and image processing, and pharmacokinetics are some areas where singularly perturbed problems arise. One can find theoretical considerations regarding singular perturbation problems in [6,9,14,23,26,29] and various approximation methods in [4,5,7,12,13,16,17,30].…”

“…In recent years various methods were employed to obtain approximations to the solution of this kind of problems. In [2,11], finite difference methods (FDM) for singularly perturbed convection-diffusion systems, in [18,19] finite element method (FEM) and in [6,13,21,23] finite difference * Correspondence: suleyman.cengizci@antalya.edu.tr 2010 AMS Mathematics Subject Classification: 65L10, 65L11 methods for singularly perturbed reaction-diffusion systems are examined. For singularly perturbed system of reaction-diffusion BVPs, in [22], the authors provided a robust computational technique , whereas in [9], optimal order error estimates are obtained on equidistributed grids.…”