Solution of Boundary Value Problems by Approaching Spline Techniques

P Kalyani,P S Rama Chandra Rao

doi:10.1155/2013/482050

P Kalyani, P S Rama Chandra Rao

Open Access

https://doi.org/10.1155/2013/482050

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Abstract

In the present work a nonpolynomial spline function is used to approximate the solution of the second order two point boundary value problems. The classes of numerical methods of second order, for a specific choice of parameters involved in nonpolynomial spline, have been developed. Numerical examples are presented to illustrate the applications of this method. The solutions of these examples are found at the nodal points with various step sizes and with various parameters (α, β). The absolute errors in each example are estimated, and the comparison of approximate values, exact values, and absolute errors of at the nodal points are shown graphically. Further, shown that nonpolynomial spline produces accurate results in comparison with the results obtained by the B-spline method and finite difference method.

Highlights

There are many linear and nonlinear problems in science and engineering, namely, second order differential equations with various types of boundary conditions, which are solved either analytically or numerically
Chawla and Katti [5] employed finite difference method for a class of singular two-point boundary value problems (BVPs); a class of BVPs was solved by using numerical integration [6]; Ravi kanth and Reddy dealt with cubic spline [7]; the variational iteration method was proposed originally by He [8] in 1999; Adomian et al solved a generalization of Airy’s equation by decomposition method [9]
The spline function we propose has the following form: T3 = Span{1, x, cos τx, sin τx}, where τ is the frequency of the trigonometric part of the spline function which can be real or pure imaginary and which will be used to raise the accuracy of the method

Summary

Introduction

There are many linear and nonlinear problems in science and engineering, namely, second order differential equations with various types of boundary conditions, which are solved either analytically or numerically. In the present communication we apply nonpolynomial spline functions to develop numerical method for obtaining the approximations to the solution of second order two point boundary value problem of the form. This type of problem (by missing the term containing u(x)) is proposed by the authors in [10, 11].

Description of the Method

Numerical Illustrations

Conclusions