NUMERICAL METHOD FOR SOLVING SINGULAR PERTURBED PROBLEM

Abstract

Mathematical equations are used to model a variety of phenomena in natural sciences and engineering. Various parameters are included in these mathematical equations. The solutions of these equations are affected by minor changes in these parame ters. This minor alteration is called perturbation and the corresponding parameter is known as the perturbation parameter. It is difficult to find the exact solutions of these mathematical equations. Therefore, the alternative way is to find their approximate solutions. These solutions are ob tained by using the approximation techniques. These Perturbation techniques further pave the way to Perturbation theory. We begin with perturbation theory in chemical kinetics. With the introduction of Michaelis-Menten mechanism and steady state approximation the concept of singular perturbation theory in chemical kinetics is studied. As we move further, we discuss a weakly coupled system of m-equations and study a highly significant numerical method i.e. q-stage runge Kutta method. We then discuss a number of iterative methods to solve initial- and/or boundary-value problems in ordinary and partial differential equations. As a series of iterates, these iterative procedures have the solution or a close approximation to it. We present and evaluate an iterative analytic approach based on the Lagrange multiplier technique to estimate the multiscale solution. Iteration is used to achieve closed-form analytic approximations to nonlinear bound ary value problems. In a general setting, variational theory and Liouville–Green transforms are used to obtain the Lagrange multiplier optimally. We have taken singular peturbed problem to test the method and also compare it with the exact so lution. Further, two test partial differential equations problems are taken into account and the findings of a detailed comparative study are discussed.