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.
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 where is an arbitrary function that is sufficiently differentiable. Expanding the test functions and about and collecting the terms we obtain whose coefficients for are constants and given as According to Jator [2], the linear multistep method is said to be of order if The main method (8) and the additional methods (9) are the order eight methods with the error constants; are , , , and , respectively. With the order , 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 with
Since the roots of (18) satisfy for , 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]): and theoretical solution: .
Problem 2. Consider the nonlinear fourth order problem (see [9]): and theoretical solution: .
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].
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 passes along a ship or offshore structure, the resultant fluid actions vary with time . In a particular case study by Wu et al. [5], the fourth order problem is defined as which is subjected to the following initial conditions: where for the existence of the theoretical solution, . The theoretical solution is undefined when (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 and with . 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 at the end point . BHCM4 manages to achieve better accuracy compared to Adams Bashforth-Adams Moulton method, Twizell [6], and Cortell [7] when and , respectively.
Figure 1 depicts the numerical solution for Wu equation (22) with and in the interval . The solutions obtained by BHCM4 are in agreement with the observation of Cortell [7] and Mathematica built-in package NDSolve.
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 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.