SOME RECENT APPROXIMATE SOLUTION TO NON LINEAR BOUNDARY VALUE PROBLEMS (STATISTICS PROJECT TOPICS AND MATERIALS)
ABSTRACT
This paper features a survey of some recent techniques for solving some nonlinear boundary value problems. Galerkin method is applied to some problems while Variational Iteration Method (VIM) and Homotopy Perturbation Method (HPM) are introduced later. The results of some comparison of these methods are given in the thesis.
CHAPTER ONE
GENERAL INTRODUTION
1.1 BACKGROUND TO THE STUDY
Non-linear phenomena play a crucial role in applied mathematics and engineering. In the previous years, so many mathematical methods that are aimed at obtaining analytical solutions of non-linear boundary value problems arising in various fields of science and engineering have been introduced and used.
 However, most of them require a tedious analysis or a large computer memory to handle these problems.
 In this paper we present and compare some methods which are recently studied by the scientists to obtain approximate analytical solutions of some nonlinear boundary value problems arising in various fields of science and engineering.
The first method considered in this research is the Galerkin Method which was introduced by Boris Galerkin in 1915 [1, 2]. Galerkin Method as an approximating solution has been shown to be an effective technique from both theoretical and practical point of view for approximating the solution to linear and mildly non-linear boundary value problems [3].
Then, the Variational Iteration Method (VIM) which is based on the incorporation of a general Lagrange multiplier in the construction of correction  functional for the equation. This method has been proposed by Shou and He [4] and is thoroughly used by many researchers  [5, 6] to handle linear and non-linear problems. The VIM uses only the prescribed conditions and does not require a specific treatment. The VIM is capable of solving a large class of linear or non-linear differential equations without the tangible restriction of sensitivity to the degree of the non-linear term and also it reduces the size of calculations.
 The homotopy perturbation method (HPM) which was proposed by  He [7] in 1999 has the solution obtain as the summation of an infinite series, which converges to analytical solution. Using the homotopy technique from topology, a homotopy is constructed with an embedding parameter , which is considered as a “small parameter“. The approximations obtained by the HPM are uniformly valid not only for small parameters but also for very large parameters. Also, this method is modified and used by some scientists to obtain a fast convergent rate [8].
