Mathematical Problems in Engineering

Volume 2015, Article ID 561489, 6 pages

http://dx.doi.org/10.1155/2015/561489

## Block Hybrid Collocation Method with Application to Fourth Order Differential Equations

^{1}Department of Mathematical and Actuarial Sciences, Universiti Tunku Abdul Rahman, Setapak, 53300 Kuala Lumpur, Malaysia^{2}Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Received 10 July 2014; Revised 22 November 2014; Accepted 25 November 2014

Academic Editor: María Isabel Herreros

Copyright © 2015 Lee Ken Yap and Fudziah Ismail. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The block hybrid collocation method with three off-step points is proposed for the direct solution of fourth order ordinary differential equations. The interpolation and collocation techniques are applied on basic polynomial to generate the main and additional methods. These methods are implemented in block form to obtain the approximation at seven points simultaneously. Numerical experiments are conducted to illustrate the efficiency of the method. The method is also applied to solve the fourth order problem from ship dynamics.

#### 1. Introduction

Fourth order ordinary differential equations (ODEs) arise in several fields such as fluid dynamics (see [1]), beam theory (see [2, 3]), electric circuits (see [4]), ship dynamics (see [5–7]), and neural networks (see [8]). Therefore, many theoretical and numerical studies dealing with such equations have appeared in the literature.

Here, we consider general fourth order ordinary differential equations: with the initial conditions Conventionally, fourth order problems (1) are reduced to system of first order ODEs and solved with the methods available in the literature. Many investigators [2, 9, 10] remarked the drawback of this approach as it requires heavier computational work and longer execution time. Thus, the direct approach on higher order ODEs has attracted considerable attention.

Recent developments have led to the implementation of collocation method for the direct solution of fourth order ODEs (1). Awoyemi [9] proposed a multiderivative collocation method to obtain the approximation of at . Moreover, Kayode [11, 12] developed collocation methods for the approximation of at with the predictor of orders five and six, respectively. These schemes [9, 11, 12] are implemented in predictor-corrector mode with the employment of Taylor series expansions for the computation of starting values. Jator [2] remarked that the implementation of these schemes is more costly since the subroutines for incorporating the starting values lead to lengthy computational time. Thus, some attempts have been made on the self-starting collocation method which eliminates the requirement of either predictors or starting values from other methods. Jator [2] derived a collocation multistep method and used it to generate a new self-starting finite difference method. On the other hand, Olabode and Alabi [13] developed a self-starting direct block method for the approximation of at , .

Here, we are going to derive a block hybrid collocation method for the direct solution of general fourth order ODEs (1). The method is extended from the line proposed by Jator [14] and Yap et al. [15]. We apply the interpolation and collocation technique on basic polynomials to derive the main and additional methods which are combined and used as block hybrid collocation method. This method generates the approximation of at four main points and three off-step points concurrently.

#### 2. Derivation of Block Hybrid Collocation Methods

The hybrid collocation method that generates the approximations to the general fourth order ODEs (1) is defined as follows: We approximate the solution by considering the interpolating function where , are unknown coefficients to be determined, is the number of interpolations for , and is the number of distinct collocation points with . The continuous approximation is constructed by imposing the conditions as follows: where , , and are not integers. By considering and , we interpolate (5) at the points , , , and and collocate (6) at the points , , , , , , , and . This leads to a system of twelve equations which is solved by Mathematica. The values of are substituted into (4) to develop the multistep method: where , , and are constant coefficients. Hence, the block hybrid collocation method can be derived as follows.

*Main Method*. Consider the following

*Additional Method*. Consider the following

The general fourth order differential equations involve the first, second, and third derivatives. In order to generate the formula for the derivatives, the values of are substituted into

This is obtained by imposing that

The formula for the first, second, and third derivatives is depicted in Tables 1, 2, and 3, respectively.