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Lee Ken Yap 1, 2 and Fudziah Ismail 2

Academic Editor:Maria Isabel Herreros

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; 22 February 2015

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.

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: [figure omitted; refer to PDF] with the initial conditions [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] at [figure omitted; refer to PDF] . Moreover, Kayode [11, 12] developed collocation methods for the approximation of [figure omitted; refer to PDF] at [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] at [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

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

Main Method . Consider the following [figure omitted; refer to PDF]

Additional Method . Consider the following [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] are substituted into [figure omitted; refer to PDF]

This is obtained by imposing that [figure omitted; refer to PDF]

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

Table 1: Coefficients [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for the method (11) evaluated at [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Table 2: Coefficients [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for the method (12) evaluated at [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Table 3: Coefficients [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for the method (13) evaluated at [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

3. Order and Stability Properties

Following the idea of Henrici [16] and Jator [2, 14], the linear difference operator associated with (3) is defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is an arbitrary function that is sufficiently differentiable. Expanding the test functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] about [figure omitted; refer to PDF] and collecting the terms we obtain [figure omitted; refer to PDF] whose coefficients [figure omitted; refer to PDF] for [figure omitted; refer to PDF] are constants and given as [figure omitted; refer to PDF] According to Jator [2], the linear multistep method is said to be of order [figure omitted; refer to PDF] if [figure omitted; refer to PDF] The main method (8) and the additional methods (9) are the order eight methods with the error constants; [figure omitted; refer to PDF] are [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively. With the order [figure omitted; refer to PDF] , we stipulate the consistency of the method (see [2, 16]).

In the sense of Jator [2], the hybrid methods (8)-(9) are normalized in block form to analyze the zero stability. The first characteristic polynomial is defined as [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

Since the roots of (18) satisfy [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , the method is zero stable.

4. Numerical Experiment

The following problems are solved numerically to illustrate the efficiency of the block hybrid collocation method.

Problem 1.

Consider the linear fourth order problem (see [2]): [figure omitted; refer to PDF] and theoretical solution: [figure omitted; refer to PDF] .

Problem 2.

Consider the nonlinear fourth order problem (see [9]): [figure omitted; refer to PDF] and theoretical solution: [figure omitted; refer to PDF] .

The block hybrid collocation method is implemented together with the Mathematica built-in packages, namely, Solve and FindRoot for the solution of linear and nonlinear problems, respectively.

The performance comparison between block hybrid collocation method with the existing methods [2, 9] and the Adams Bashforth-Adams Moulton method is presented in Tables 4 and 5. The following notations are used in the tables:

: h : step size;

: BHCM4: block hybrid collocation method;

: Adams: Adams Bashforth-Adams Moulton method;

: Awoyemi: multiderivative collocation method in Awoyemi [9];

: Jator: finite difference method in Jator [2].

Table 4: Numerical results for Problem 1.

[figure omitted; refer to PDF]	Method	Absolute error at [figure omitted; refer to PDF]
0.1	BHCM4	1.74 (-8)
Adams	2.11 (-3)
Jator	1.26 (-4)

0.05	BHCM4	8.45 (-11)
Adams	5.37 (-4)
Jator	1.91 (-6)

0.025	BHCM4	3.69 (-13)
Adams	5.09 (-5)
Jator	2.96 (-8)

0.02	BHCM4	7.11 (-14)
Adams	2.25 (-5)
Jator	8.65 (-9)

Table 5: Numerical results for Problem 2.

[figure omitted; refer to PDF]	Method	Absolute error at [figure omitted; refer to PDF]
0.2	BHCM4	2.38 (-12)
Adams	5.01 (-7)
Awoyemi	5.84 (-4)

0.1	BHCM4	1.95 (-14)
Adams	2.44 (-6)
Awoyemi	9.26 (-5)

Tables 4 and 5 show the superiority of BHCM4 in terms of accuracy over the existing Adams method, Jator finite difference method [2], and Awoyemi multiderivative collocation method [9].

5. Application to Problem from Ship Dynamics [5-7]

The proposed method is also applied to solve a physical problem from ship dynamics. As stated by Wu et al. [5], when a sinusoidal wave of frequency [figure omitted; refer to PDF] passes along a ship or offshore structure, the resultant fluid actions vary with time [figure omitted; refer to PDF] . In a particular case study by Wu et al. [5], the fourth order problem is defined as [figure omitted; refer to PDF] which is subjected to the following initial conditions: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] for the existence of the theoretical solution, [figure omitted; refer to PDF] . The theoretical solution is undefined when [figure omitted; refer to PDF] (see [6]).

In the literature, some numerical methods for solving fourth order ODEs have been extended to solve the problem from ship dynamics. Numerical investigation was presented in Twizell [6] and Cortell [7] concerning the fourth order ODEs (22) for the cases [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . Instead of solving the fourth order ODEs directly, Twizell [6] and Cortell [7] considered the conventional approach of reduction to system of first order ODEs. Twizell [6] developed a family of numerical methods with the global extrapolation to increase the order of the methods. On the other hand, Cortell [7] proposed the extension of the classical Runge-Kutta method.

Table 6 shows the comparison in terms of accuracy for [figure omitted; refer to PDF] at the end point [figure omitted; refer to PDF] . BHCM4 manages to achieve better accuracy compared to Adams Bashforth-Adams Moulton method, Twizell [6], and Cortell [7] when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively.

Table 6: Performance comparison for Wu equation with [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	Method	Absolute error at [figure omitted; refer to PDF]
0.25	BHCM4	5.2 (-7)
Adams	4.9 (-3)
Twizell	1.9 (-4)

0.1	BHCM4	2.8 (-10)
Adams	8.4 (-5)
Cortell	3.7 (-5)

Figure 1 depicts the numerical solution for Wu equation (22) with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in the interval [figure omitted; refer to PDF] . The solutions obtained by BHCM4 are in agreement with the observation of Cortell [7] and Mathematica built-in package NDSolve.

Figure 1: Response curve for Wu equation with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

6. Conclusion

As indicated in the numerical results, the block hybrid collocation method has significant improvement over the existing methods. Furthermore, it is applicable for the solution of physical problem from ship dynamics.

As a conclusion, the block hybrid collocation method is proposed for the direct solution of general fourth order ODEs whereby it is implemented as self-starting method that generates the solution of [figure omitted; refer to PDF] at four main points and three off-step points concurrently.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

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[3] O. Kelesoglu, "The solution of fourth order boundary value problem arising out of the beam-column theory using Adomian decomposition method," Mathematical Problems in Engineering , vol. 2014, 2014.

[4] A. Boutayeb, A. Chetouani, "A mini-review of numerical methods for high-order problems," International Journal of Computer Mathematics , vol. 84, no. 4, pp. 563-579, 2007.

[5] X. J. Wu, Y. Wang, W. G. Price, "Multiple resonances, responses, and parametric instabilities in offshore structures," Journal of Ship Research , vol. 32, no. 4, pp. 285-296, 1988.

[6] E. H. Twizell, "A family of numerical methods for the solution of high-order general initial value problems," Computer Methods in Applied Mechanics and Engineering , vol. 67, no. 1, pp. 15-25, 1988.

[7] R. Cortell, "Application of the fourth-order Runge-Kutta method for the solution of high-order general initial value problems," Computers and Structures , vol. 49, no. 5, pp. 897-900, 1993.

[8] A. Malek, R. Shekari Beidokhti, "Numerical solution for high order differential equations using a hybrid neural network-optimization method," Applied Mathematics and Computation , vol. 183, no. 1, pp. 260-271, 2006.

[9] D. O. Awoyemi, "Algorithmic collocation approach for direct solution of fourth-order initial-value problems of ordinary differential equations," International Journal of Computer Mathematics , vol. 82, no. 3, pp. 321-329, 2005.

[10] N. Waeleh, Z. A. Majid, F. Ismail, M. Suleiman, "Numerical solution of higher order ordinary differential equations by direct block code," Journal of Mathematics and Statistics , vol. 8, no. 1, pp. 77-81, 2011.

[11] S. J. Kayode, "An order six zero-stable method for direct solution of fourth order ordinary differential equations," American Journal of Applied Sciences , vol. 5, no. 11, pp. 1461-1466, 2008.

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[13] B. T. Olabode, T. J. Alabi, "Direct block predictor-corrector method for the solution of general fourth order ODEs," Journal of Mathematics Research , vol. 5, no. 1, pp. 26-33, 2013.

[14] S. N. Jator, "Solving second order initial value problems by a hybrid multistep method without predictors," Applied Mathematics and Computation , vol. 217, no. 8, pp. 4036-4046, 2010.

[15] L. K. Yap, F. Ismail, N. Senu, "An accurate block hybrid collocation method for third order ordinary differential equations," Journal of Applied Mathematics , vol. 2014, 2014.

[16] P. Henrici Discrete Variable Methods in Ordinary Differential Equations , John Wiley & Sons, New York, NY, USA, 1962.

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Copyright © 2015 Lee Ken Yap and Fudziah Ismail. Lee Ken Yap et al. 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.

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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.

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Title

Block Hybrid Collocation Method with Application to Fourth Order Differential Equations

Author

Lee Ken Yap; Ismail, Fudziah

Publication year

2015

Publication date

2015

Publisher

John Wiley & Sons, Inc.

ISSN

1024123X

e-ISSN

15635147

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1155/2015/561489

ProQuest document ID

1661306823

Block Hybrid Collocation Method with Application to Fourth Order Differential Equations

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