## SOME RECENT APPROXIMATE SOLUTION TO NON LINEAR BOUNDARY VALUE PROBLEMS

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.

Title page
Abstract

Chapter One
General Introduction
1.1      Background to the Study
1.2      Aim/Objective of the work
1.3      Galerkin Methods
1.4      Definition of some term
1.5      Variational Iteration Methods (VIM)
1.6      Perturbation Methods

Chapter Two
Classical Methods of Solving Boundary Value Problems (BVPs)
2.1      Boundary Value Problems (BVPs)
2.2      Types of BVPs
2.3      Some Classical Methods of Solving BVPs

Chapter Three
Some Recent Methods of Solving Non-Linear BVPs
3.1      Non- Linear BVPs
3.2      Applications of VIM to Non-Linear BVPs
3.3      Applications of Classical Perturbation Methods to Non-Linear BVPs
3.4      Applications of Homotopy Perturbation Methods (HPM) to Non-Linear BVPs

Chapter Four
Applications of the Recent Methods to Buckling Problems
4.1      Approximate Solution of Classical Buckling State
4.2      Solving for an Approximate Solution of Classical Buckling State using Galerkin Methods
4.3      Solving for an Approximate Solution of Classical Buckling State using VIM
4.4      Solving for an Approximate Solution of Classical Buckling State using HPM
4.5      Post Buckling or Primary Buckled State
4.6      Solving for an Approximate Solution of Post Buckling State using VIM
4.7      Solving for an Approximate Solution of Post Buckling State using HPM

Chapter Five
Numerical Result and Conclusion
5.1 Numerical Results
5.2 Conclusion

Reference

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].
In   [9], the author features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly non-linear equations but also for strongly ones. The limitations of the traditional perturbation methods are illustrated, various modified perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In   [10], the author pays particular attention throughout the paper to give an intuitive grasp for Lagrange multiplier, calculus of variations, optimization, VIM, parameter-expansion method, exp-function method, HPM, and ancient Chinese mathematics as well. Subsequently, nanomechanics in textile engineering and E-infinity theory in high-energy physics, Kleiber's 3/4 law in biology, possible mechanism in spider-spinning process, and fractal approach to carbon nanotube are briefly introduced. In   [11], the same author presents a coupling method of a homotopy technique and a perturbation technique to solve non-linear problems. In contrast to traditional perturbation methods, HPM does not require a small parameter in the equation.
In this research work, we use these methods to solve the Classical buckling and Post or Primary buckling problem from the secondary bifurcation and imperfection sensitivity of Columns of Nonlinear Foundation [18].

1.2 AIM/OBJECTIVE OF THE WORK
In this research work, we use Galerkin method, Variational Iteration Method and Homotopy Perturbation Method comparing with the exact solution of  the classical buckling and post or primary buckling problem from the secondary bifurcation and imperfection sensitivity of columns of nonlinear foundation to check which is the better than others.

1.3 GALERKIN METHOD

The analysis of the Galerkin method proceeds in two steps. First we will show that the Galerkin equation is a well-posed problem in the sense of Hadamard and therefore admits a unique solution. Secondly we study the quality of approximation of the Galerkin solution . If we restrict the analysis to bilinear form that is.....

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