Abstract
The block hybrid collocation method with two off-step points is proposed for the direct solution of general third order ordinary differential equations. Both the main and additional methods are derived via interpolation and collocation of the basic polynomial. These methods are applied in block form to provide the approximation at five points concurrently. The stability properties of the block method are investigated. Some numerical examples are tested to illustrate the efficiency of the method. The block hybrid collocation method is also implemented to solve the nonlinear Genesio equation and the problem in thin film flow.
1. Introduction
Consider the general third order ordinary differential equations (ODEs): with the initial conditions
In particular, the third order differential equations arise in many physical problems such as electromagnetic waves, thin film flow, and gravity-driven flows (see [1–6]). Therefore, third order ODEs have attracted considerable attention. Many theoretical and numerical studies dealing with such equations have appeared in the literature. The popular approach for solving third order ODEs is by converting the problems to a system of first order ODEs and solving it using the method available in the literature. Awoyemi and Idowu [7], Jator [8], Mehrkanoon [9], and Bhrawy and Abd-Elhameed [10] remarked the drawback of this approach whereby it required complicated computational work and lengthy execution time. The studies on direct approach to higher order ODEs demonstrated the advantages in speed and accuracy.
Some attentions [8, 11–14] have been focused on direct solution of second order ODEs. Fatunla [12] suggested the zero-stable 2-point block method to solve special second order ODEs. On the other hand, Omar et al. [13] and Majid and Suleiman [14] studied parallel implementation of the direct block methods. Jator [8, 11] proposed a class of hybrid collocation methods and emphasized the accuracy advantage on self-starting method. The only necessary starting value for evaluation at the next block is the last value from the previous block. Since the loss of accuracy does not affect the subsequent points, the order of the method is maintained.
Some attempts have been made to solve third order ODEs directly using collocation method. Awoyemi [15] considered the P-stable linear multistep collocation method. Meanwhile, Awoyemi and Idowu [7] proposed the hybrid collocation method with an off-step point, . Both schemes are implemented in predictor-corrector mode to obtain the approximation at . The Taylor series expansion is employed for the computation of initial values. Olabode and Yusuph [16] applied the interpolation and collocation technique on power series to derive 3-step block method, and it was implemented as simultaneous integrator to special third order ODEs. Bhrawy and Abd-Elhameed [10] developed the shifted Jacobi-Gauss collocation spectral method for general nonlinear third order differential equations. Adesanya et al. [17] proposed a self-starting block predictor-corrector method whereby the derivation involved interpolation and collocation of power series at , for and , respectively.
Several direct variable step methods have also been proposed in literature to solve general third order ODEs. For instance, Mehrkanoon [9] implemented the direct three-point block multistep method of Adams type formulas in PECE mode with variable step size and Gauss Seidel iteration. Majid et al. [18] presented the 2-point 4-step implicit block method with the application of the simple form in Adams-Moulton method using variable step size.
Here, we are going to derive the block hybrid collocation method for the direct solution of general third order ODEs. The method is along the lines proposed by Jator [11] and Awoyemi and Idowu [7]. The derivation involves interpolation and collocation of the basic polynomial. The collocation method approximates the solution of with basic polynomial which satisfies the initial conditions and differential equations at all points. This approach generates the main and the additional methods which can be combined and used as block method. In -point block method, the interval is divided into series of blocks with each block containing -points. The application of -point block method generates a block of new solution concurrently.
2. Derivation of Block Hybrid Collocation Methods
The hybrid collocation method that produces approximations , , and to the general third order ODEs is given as follows:
In order to obtain (3), we approximate the solution by the interpolating function of the form where(i),(ii) are unknown coefficients to be determined,(iii) is the number of interpolations for , and(iv) is the number of distinct collocation points with .
The continuous approximation is constructed by imposing the following conditions: where and are not integers. Interpolating (5) at the points , , and collocating (6) at the points , , , , , and lead to a system of nine equations, which can be solved by Mathematica software to obtain the coefficient . The values of are substituted into (4) to obtain the continuous multistep method of the form where , , and are constant coefficients. Hence, the block hybrid collocation method can be derived as follows.
Main Method. Consider
Additional Method. Consider
It is noted that the general third order ODEs involve the first and second derivatives. These derivatives can be obtained by imposing that
The values of are substituted into to generate the formula for the first and second derivatives of the method. Thus, we obtain
3. Order and Stability Properties
Extending the idea of Henrici [19] and Jator [8], the linear difference operator associated with (3) is defined by where is an arbitrary function that is sufficiently differentiable. We expand the test function and its third derivative about and collect the terms to obtain whose coefficients for are constants and given as
The associated linear multistep method and the linear difference operator are said to be of order , if and . The main method and the two additional methods have order with the error constants; are , , and , respectively. The block hybrid collocation method is consistent since it has order .
To analyze the zero stability, (8)–(12) are normalized. Zero stability can be described by matrix finite difference equation as follows: with and constant matrices
The first characteristic polynomial is defined as
Since the roots of the first characteristic polynomial are modulus less than or equal to one, the method is zero stable. This property, together with the consistency of the method, implies the convergence (see Henrici [19]) of the block method.
4. Numerical Examples and Discussion
To illustrate the effectiveness of the block hybrid collocation method, the test Problems 1–3 are solved numerically. The block hybrid collocation method is implemented and leads to a system of fifteen equations. The code is written and executed in Mathematica 8.0 to obtain the numerical solutions. A built-in Mathematica package, namely, LinearSolve, is applied in the algorithm to solve the linear system for numerical results. In general, this package solves the matrix equation and returns a vector .
The block hybrid collocation method is then compared with the existing methods [7, 15–17, 20] for direct solution of general third order ODEs. The block hybrid collocation method is also compared with the Adams Bashforth-Adams Moulton method. The well-known fourth order Runge Kutta method is applied to obtain the starting values for Adams method, whereby the third order ODEs are reduced to system of first order ODEs.
Problem 1. Consider
Exact Solution: .
Source: Majid et al. [18].
Problem 2. Consider
Exact Solution: .
Source: Majid et al. [18].
Problem 3. Consider
Exact Solution: .
Source: Adesanya et al. [17].
The performance comparison between block hybrid collocation method with the existing results [7, 9, 15–17, 20] and the Adams Bashforth-Adams Moulton method is presented in Tables 1–4. The following notations are used in the tables. : step size. BHCM: block hybrid collocation method. Adams: Adams Bashforth-Adams Moulton method. Awoyemi (1): P-stable multistep method in Awoyemi [15]. Awoyemi (2): hybrid collocation method in Awoyemi and Idowu [7]. Awoyemi (3): nonsymmetric collocation method in Awoyemi et al. [20]. Adesanya: block predictor-corrector method in Adesanya et al. [17]. Mehrkanoon: variable step three-point block multistep method in Mehrkanoon [9]. Olabode: block method for special third order ODEs in Olabode and Yusuph [16]. Step: total number of steps taken to obtain solution. Time: execution time taken in microseconds. Maxe: magnitude of the maximum error of the computed solution.
The maximum error is defined as
Tables 1–3 show that the block hybrid collocation method has better performance in terms of accuracy and number of steps to obtain the solution compared to Adams Bashforth-Adams Moulton method.
In Table 1, a direct comparison is made between Awoyemi methods [7, 15] and our block hybrid collocation method. For , BHCM has better approximation compared to Awoyemi hybrid collocation method [7] for step sizes and . Even in the larger interval (), BHCM has higher order of accuracy when compared to Awoyemi P-stable multistep method [15] using and . It is obvious that BHCM requires less steps to obtain the solutions compared to Awoyemi methods [7, 15].
Table 2 shows that BHCM manages to achieve better accuracy and less total steps compared to Awoyemi P-stable multistep method [15] for constant step size when , and . Furthermore, BHCM gain better accuracy compared to Awoyemi nonsymmetric collocation method [20] and Adesanya block predictor-corrector method [17] when and . Table 3 shows the superiority of BHCM in terms of accuracy over Adesanya block predictor-corrector method [17] and Olabode block method [16] for special third order ODEs.
Figures 1, 2, and 3 depict the performance comparison between BHCM and the existing methods: Awoyemi hybrid collocation method [7], Awoyemi P-stable multistep method [15], and Adams Bashforth-Adams Moulton method, respectively. It shows that BHCM is more efficient than the existing methods in terms of accuracy. BHCM also requires less computational time compared to the existing methods. This is expected since BHCM calculates the approximation at three main points concurrently.
(a) Maxe versus step size
(b) Time versus step size
5. Application to Solve Nonlinear Genesio Equation
Here we consider the nonlinear chaotic system from Genesio and Tesi [1] with that is subject to the following initial conditions: where , , and are the positive constants that satisfied for the existence of the solution.
BHCM is implemented together with a built-in Mathematica package, namely, FindRoot, for the solution of nonlinear system based on Newton’s method. In order to apply BHCM and Newton’s method at the next block, the last value from previous block is used as starting value and initial guess, respectively.
The numerical solutions are compared with the numerical solutions obtained in Mehrkanoon [9], Bataineh et al. [2], and the Mathematica built-in package NDSolve.
Table 4 shows the comparison in the numerical approximation of at the end points and . BHCM achieves similar approximation as Mehrkanoon variable step three-point block multistep method [9].
Figure 4 depicts the numerical solutions for the nonlinear Genesio equation [1] in the interval . Bataineh et al. [2] stated that NHAM is more stable than the numerical solution obtained by classical HAM. In fact, the solutions obtained by BHCM are in agreement with the observation of Bataineh et al. [2] using NHAM and Mathematica built-in package NDSolve.
![(a) Solutions obtained by seven- and eight-order Runge Kutta method (RK78), homotopy analysis method (HAM), and new variant of HAM (NHAM) (adapted from Bataineh et al. [2])](https://static.hindawi.com/articles/jam/volume-2014/549597/figures/549597.fig.004a.jpg)
(a) Solutions obtained by seven- and eight-order Runge Kutta method (RK78), homotopy analysis method (HAM), and new variant of HAM (NHAM) (adapted from Bataineh et al. [2])
| (b) Solutions obtained by BHCM with |
6. Application to Solve Problem in Thin Film Flow
The proposed method is also applied to solve the well-known physical problem regarding the thin film flow of a liquid. In fluid dynamics, Tuck and Schwartz [3] investigated some third order ODEs that are relevant to draining and coating flows. They discussed the motion of the fluid on a plane surface in which the flow is in the direction of motion along the plane. This fluid dynamics problem was formulated in an autonomous third order ODEs where
In the literature, some numerical methods for solving special third order ODEs have been extended to solve the problem in thin film flow. Some numerical investigation was presented in Momoniat and Mahomed [4] and Mechee et al. [5] concerning the special third order ODEs with the initial conditions for the cases and .
Instead of using the conventional approach of reduction to system of first order ODEs, Momoniat and Mahomed [4] applied the successive reduction by writing the third order ODEs (29) in terms of the differential invariants and solved it by fourth order Runge Kutta method. Mechee et al. [5] applied the three-stage fifth order Runge Kutta method to solve the third order physical problems (29) directly.
Here we will apply the proposed block hybrid collocation method (BHCM) to solve the third order ODEs (29). The numerical solutions are compared with Momoniat and Mahomed [4] and Mechee et al. [5] in the literature. The results are displayed in Tables 5–8.
Table 5 demonstrates that BHCM performs slightly better compared to Mechee et al. [5] with for the case . In Table 6, we observe that Momoniat and Mahomed [4], Mechee et al. [5], and BHCM have similar order of accuracy.
Tables 7 and 8 show the numerical results for the case with and , respectively. In fact, the case cannot be solved analytically. Table 7 shows that BHCM manages to achieve the numerical result which agrees to six decimal places when compared to Mechee et al. [5] for . In Table 8, the numerical results for BHCM agree to nine decimal places when compared to Momoniat and Mohamed [4], while they agree to ten decimal places when compared to Mechee et al. [5] for .
Figure 5 shows the total number of steps taken to obtain the solutions versus method. It is obvious that BHCM require less steps to obtain the solution compared to Mechee et al. [5] and Momoniat and Mahomed [4]. The results are expected since the BHCM calculates the values of at 3 points simultaneously while the Runge Kutta method in Mechee et al. [5] and Momoniat and Mahomed [4] calculates only one value of at a time.
| (a) Total steps when |
| (b) Total steps when |
BHCM is clearly superior in solving the problem in thin film flow (29) since it involves less computational work and yields highly accurate solutions.
7. Conclusion
As a whole, the numerical results demonstrate the efficiency of the block hybrid collocation method. It is observed that the block hybrid collocation method requires less number of total steps compared to Awoyemi methods [7, 15, 20]. It reduces the total number of steps to almost one-third to obtain the solution. These results are expected since the -point block methods calculate the values of at three main points concurrently.
The numerical results also demonstrate the accuracy of the block hybrid collocation method. It gives precise approximation as the step size decreases. When compared with the existing methods [7, 15–17, 20] and Adams Bashforth-Adams Moulton method, the block hybrid collocation method meets better accuracy for various step sizes even in larger interval.
As a conclusion, the -point block hybrid collocation method with two off-step points has been proposed and implemented as a self-starting method for third order ordinary differential equations. The results suggest a significant improvement in efficiency of the block hybrid collocation method in the direct solution of general third order ODEs. The block hybrid collocation method is applicable to solve the nonlinear Genesio equation [1] and the physical problem in thin film flow [4, 5].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.