Partial Differential Equations and Solitary Waves Theory

"Part I Partial Differential Equations (p. 1-3)

Chapter 1 Basic Concepts

1.1 Introduction

It is well known that most of the phenomena that arise in mathematical physics and engineering fields can be described by partial differential equations (PDEs). In physics for example, the heat flow and the wave propagation phenomena are well described by partial differential equations [1–4]. In ecology, most population models are governed by partial differential equations [5–6].

The dispersion of a chemically reactive material is characterized by partial differential equations. In addition, most physical phenomena of fluid dynamics, quantum mechanics, electricity, plasma physics, propagation of shallow water waves, and many other models are controlled within its domain of validity by partial differential equations. Partial differential equations have become a useful tool for describing these natural phenomena of science and engineering models.

Therefore, it becomes increasingly important to be familiar with all traditional and recently developed methods for solving partial differential equations, and the implementation of these methods. However, in this text, we will restrict our analysis to solve partial differential equations along with the given conditions that characterize the initial conditions and the boundary conditions of the dependent variable [7].We fill focus our concern on deriving solutions to PDEs and not on the derivation of these equations.

In this text, our presentation will be based on applying the recent developments in this field and on applying some of the traditional methods as well. The formulation of partial differential equations and the scientific interpretation of the models will not be discussed. It is to be noted that several methods are usually used in solving PDEs.

The newly developed Adomian decomposition method and the related improvements of the modified technique and the noise terms phenomena will be effectively used. Moreover, the variational iteration method that was recently developed will be used as well. The recently developed techniques have been proved to be reliable, accurate and effective in both the analytic and the numerical purposes.

The Adomian decomposition method and the variational iteration method were formally proved to provide the solution in terms of a rapid convergent infinite series that may yield the exact solution in many cases. As will be seen in part I of this text, both methods require the use of conditions such as initial conditions. The other related modifications were shown to be powerful in that it accelerate the rapid convergence of the solution. However, some of the traditional methods, such as the separation of variables method and the method of characteristics will be applied as well.

Moreover, the other techniques, such as integral transforms, perturbation methods, numerical methods and other traditional methods, that are usually used in other texts, will not be used in this text. In Part II of this text, we will focus our work on nonlinear evolution equations that describe a variety of physical phenomena. The Hirota’s bilinear formalism and the tanh-coth method will be employed in the second part. These methods will be used to determine soliton solutions andmultiple-soliton solutions, for completely integrable equations, as well. Several well-known nonlinear evolution equations such as the KdV equation, Burgers equation, Boussinesq equation, Camassa-Holm equation, sine-Gordon equation, and many others will be investigated."