An Efficient Two-Step Symmetric Hybrid Block Method for Solving Second-Order Initial Value Problems of Ordinary Differential Equations
- December 12, 2019
- Posted by: RSIS
- Categories: IJRSI, Mathematics
International Journal of Research and Scientific Innovation (IJRSI) | Volume VI, Issue XI, November 2019 | ISSN 2321–2705
F.O. Obarhua
Department of Mathematical Sciences, Federal University of Technology, P. M. B. 704, Akure, Nigeria
Abstract:- A Linear Multistep Method of order six with two off-grid points is presented for direct numerical integration of second order initial value problems of ordinary differential equations. Several methods are developed using interpolation and collocation approach with special cognizance of two hybrid points which are selected to enhance the accuracy of the block methods. The properties and convergence of the proposed method are discussed. Superiority of the method over existing methods is established by implementing the method on different test problems.
Keywords: Symmetric, Hybrid method, Initial value Problems, Block method
I. BACKGROUND
In this work, initial value second-order problems of the form:
y”=f(t,y,y’), y(a)= w<sub>0</sub>, y'(a)= w<sub>1</sub> (1)
is numerically integrated where a, b, w<sub>0</sub>, w<sub>1</sub> are real numbers.
The mathematical models in engineering and many spheres of human endeavors often lead to initial value problem of ordinary differential equations (1)
Several numerical methods have been designed and proposed in literature for solving second order ordinary differential equations. For example, [1] developed a self-starting linear multistep method and applied it to solve second order IVPs of ODEs directly. Two intra step grid points were considered by means of collocation and interpolation approach. [2] proposed a single-step hybrid block method of order five to solve second order ODEs. In the work, three off-step points were approximated by collocation approach. In the work by [3], continuous hybrid multistep method with Legendre polynomial as the approximate solutions was investigated to obtain the approximation of stiff second order ODEs. Also, two intra step grid points were considered by means of collocation and interpolation approach. Moreso, [4] developed numerical solution of stiff and oscillatory first order differential equations, using the combination of power series and exponential function as basis function.
