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SOLVING BVP USING INTERPOLATION METHOD

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 Format: MS WORD ::   Chapters: 1-5 ::   Pages: 57 ::   Attributes: calculation ::   189 people found this useful
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CHAPTER ONE

INTRODUCTION

1.1 BACKGROUND OF STUDY

Boundary value problems occur very frequently in various fields of science and engineering such as mechanics, quantum physics, electro hydro dynamics, and theory of thermal expansions. For a detailed theory and analytical discussion on boundary value problems, one may refer to Bender [1], Collatz [2], Na [3],Numerical solution of boundary value problems by splines and numerical integration are discussed by P.S. Rama Chandra Rao [4-5], Ravikanth. A.S.V. [6]. The finite difference method is one of several techniques for obtaining numerical solutions to the boundary value problems. Especially to solve partial differential equations , in which the partial derivatives are replaced by finite differences of two variables .Mortan and Mayer[7] and Cooper[8] provide a more mathematical development of finite difference methods and modern introduction to the theory of partial differential equation along with a brief coverage of numerical methods. Fletcher [9] described the method to implement finite differences to solve boundary value problems.

Strongly non-linear multi-point boundary value problems (MPBVP) very often appear in a variety of problems arising in a broad discipline of sciences, such as mathematics, theoretical physics, solid and fluid mechanics, control and optimization theory etc.[1]. In this class of problems, for a set of nonlinear ordinary differential equations some boundary conditions are specified at the initial value of the independent variable and the remaining ones at its terminal value [2]. Although there exists many methods for solving MPBVPs, no precise and widely applicable methods are available to date, that can treat all different types of such problems. Furthermore, neither the existence nor the uniqueness of a solution is always guaranteed, while additional difficulties may appear due to possible singularities involved. Among the methods described in the relevant literature, the most popular ones are the shooting [3], the finite difference methods, Variational Iteration Methods [4], Homotopy Perturbation Method methods[5], Padè approximation methods, Spline Functions [6], and the recently developed method of locally transversal linearization [7].

1.2 STATEMENT OF THE PROBLEM

In a multitude of cases and for engineering problems in particular, the direct application of one of the aforementioned methods often has as a consequence the distraction from the essence of the physical problem dealt with; this occurs since most of these methods are associated with sophisticated numerical algorithms and require the decomposition of higher order differential equations into a system of 1st order ones. An efficient way to overcome this obstacle is to resort to modern commercial mathematical software and take advantage of the strong capabilities of its powerful embedded functions, a task that can be achieved without significant computational effort. In doing this, the present paper offers a simple yet efficient numerical algorithm for the solution of multi-point nonlinear boundary value problems using interpolation method.

1.3 AIM AND OBJECTIVES OF THE STUDY

The main aim of the research word is to examine solving BVP using interpolation method. The objectives of the study are:

  1. To explore various interpolation methods used in solving BVP
  2. To investigate the factors affecting the use of interpolation method for solving BVPs
  3. To determine the compare the results from the various method used in solving BVP

1.4 RESEARCH QUESTIONS

The study came up with research questions so as to ascertain the above objectives of the study. The following research questions guide the objectives of the study:

  1. What is the various interpolation methods used in solving BVP?
  2. What are the factors affecting the use of interpolation method for solving BVPs?
  3. What is the difference in the results from the various method used in solving BVP

1.5 SIGNIFICANCE OF THE STUDY

The study on solving BVP using interpolation method will be of immense benefit to the mathematics department and the tertiary institution in Nigeria. The study will look at various method used in solving BVP. The study will try to find out the level of efficiency for each method of solving BVP. The study will serve as a repository of information to other researchers that desire to carry out similar research on the above topic. Finally the study will contribute to the body of the existing literature on solving BVP using interpolation method

1.6 SCOPE OF THE STUDY

The study will focus on the shooting method interpolation method for solving BVP

1.7 DEFINITION OF TERMS

BVP: Boundary value problem

Boundary value problem: A boundary value problem is a problem, typically an ordinary differential equation or a partial differential equation, which has values assigned on the physical boundary of the domain in which the problem is specified

 


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Project Information

Format:MS WORD
Chapter:1-5
Pages:57
Attribute:calculation
Price:₦3,000
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