APPLICATION OF ADOMIAN’S DECOMPOSITION METHOD IN SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

CHAPTER ONE

INTRODUCTION

The theory of nonlinear problem has recently undergone much study. We do not attempt to characterize the general form of nonlinear equations [1]. Rather, we solve a specific equation in the following nonlinear problem by using the Adomian decomposition method [2-4]. By solving this type of problems, we do not use conventional transformations which transform a nonlinear problem to an evolution equation and the reduced to a bilinear form. Some times transformation of the nonlinear problem might produce an even more complicated problem. Nonlinear phenomena play a crucial role in applied mathematics and physics. The nonlinear problems are solved easily and elegantly without linearizing the problem by using the Adomian’s decomposition method. The Adomian decomposition method was presented in 1980’s by Adomian. The method is very useful for solving linear and nonlinear ordinary and partial differential equations, algebraic equations, functional equations, integral differential equations and the convergence analysis of the ADM was discussed in [2]. Y. Cherruault and G. Adomian give the new proof of convergence analysis of the decomposition method [16]. E. Babolian And J. Biazar, define the order of the convergence of adomian method in [11]. After that many modifications were made on this method by numerous researchers in an attempt to improve the accuracy or extend the applications of this method. In given in [9]. A new modification methods of the ADM, Wazwaz modifications and the two step modified Adomian decomposition method. In chapter 3, we will use the ADM to solve different types of differential equations. Yahya Qaid Hasan and Liu Ming Zhu modified the ADM to solve second order singular initial value ordinary differential equations [21]. Several examples on solving the ordinary differential equations, initial value problems and boundary value problems are introduced in [2]. J. Biazar, E. Babolian and R. Islam in [12] obtained the solution of a system of ordinary differential equations by using ADM. In the second part of chapter, we will apply the ADM for solving partial differential equations. We will consider first order PDEs as done by [7]. Then, we move to several 2nd order PDE’s, linear heat equation, nonlinear heat equation [14], linear wave equation and nonlinear wave equation [15]. The generalized log-KdV(Diederik Korteweg and Gustav de Vries)equation in [38] will be solved by ADM as an application on higher order PDE’s. We apply the method for solving system of PDE’s as in [7]. At the end of this chapter, we show how we can solve the integral equations by using Adomian decomposition method. In chapter 4, we review some inverse problems and show how ADM is used for solving these problems. There are many classifications of the inverse problems, we will deal with boundary conditions determination of inverse problems[31] and parameter determination for some equations[35].

TOPIC: APPLICATION OF ADOMIAN’S DECOMPOSITION METHOD IN SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

Format: MS Word

Chapters: 1 - 5, Abstract, References
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Number of Pages: 80

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