Solutions of Seventh Order Boundary Value Problems Using Ninth Degree Spline Functions and Comparison with Eighth Degree Spline Solutions

Parcha Kalyani,Mihretu Nigatu Lemma

doi:10.4236/jamp.2016.42032

Abstract

In this article, we develop numerical method by constructing ninth degree spline function using extended cubic spline Bickley’s method to find the approximate solution of seventh order linear boundary value problems at different step lengths. The approximate solution is compared with the solution obtained by eighth degree splines and exact solution. It has been observed that the approximate solution is an excellent agreement with exact solution. Low absolute error indicates that our numerical method is effective for solving high order linear boundary value problems.

Highlights

IntroductionSpline functions of various degrees have been demonstrated by them using approximate methods of solving second, third, fourth and fifth order linear boundary value problems
Consider the linear seventh order differential equation y(7) ( x) + f ( x) y ( x) = r ( x) with the boundary conditions= y ( x0 ) α=, y β=, y′( x0 ) α′=, y′( xn ) β ′, How to cite this paper: Kalyani, P. and Lemma, M.N. (2016) Solutions of Seventh Order Boundary Value Problems Using Ninth Degree Spline Functions and Comparison with Eighth Degree Spline Solutions
A ninth degree spline solution has been employed of example 1, 2 and 3 at step lengths h = 0.2 and h = 0.1

Summary

Introduction

Spline functions of various degrees have been demonstrated by them using approximate methods of solving second, third, fourth and fifth order linear boundary value problems. There are number of research articles published on this subject, yet it remains an active research area Techniques such as quadratic, cubic, quartic, quintic, sextic, septic and higher degree splines are used to discuss the numerical solution of linear and nonlinear BVPs. Kumar and Srivastava [17] have given a survey on recent spline techniques for solving boundary value problems in ordinary differential equations using cubic, quintic and sextic polynomial and non-polynomial splines. The seventh order boundary value problems are solved using ninth degree spline approximation and compared with the solution obtained by eighth degree spline solution [19]

Construction of Ninth Degree Spline

Numerical Illustrations

Example 1

Example 2

Example 3

Conclusion