An Accurate Block Hybrid Collocation Method for Third Order Ordinary Differential Equations

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 549597, 9 pages http://dx.doi.org/10.1155/2014/549597 Research Article An Accurate Block Hybrid Collocation Method for Third Order Ordinary Differential Equations Lee Ken Yap,1,2 Fudziah Ismail,2 and Norazak Senu2 1 2 Department of Mathematical and Actuarial Sciences, Universiti Tunku Abdul Rahman, Setapak, 53300 Kuala Lumpur, Malaysia Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia Correspondence should be addressed to Lee Ken Yap; Received 24 January 2014; Revised 24 April 2014; Accepted 7 May 2014; Published 27 May 2014 Academic Editor: Kai Diethelm Copyright © 2014 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. 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): 𝑦󸀠󸀠󸀠 = 𝑓 (𝑥, 𝑦, 𝑦󸀠 , 𝑦󸀠󸀠 ) , (1) with the initial conditions 𝑦 (𝑎) = 𝑦0 , 𝑦󸀠 (𝑎) = 𝑦0󸀠 , 𝑦󸀠󸀠 (𝑎) = 𝑦0󸀠󸀠 , 𝑥 ∈ [𝑎, 𝑏] . (2) 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, 𝑥𝑛+3/2 . Both schemes are implemented in predictor-corrector mode to obtain the approximation at 𝑥𝑛+3 . 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. 2 Journal of Applied Mathematics Adesanya et al. [17] proposed a self-starting block predictorcorrector method whereby the derivation involved interpolation and collocation of power series at 𝑥𝑛+𝑗 , for 𝑗 = 1(1)3 and 𝑗 = 0(1)4, 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 threepoint 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 AdamsMoulton 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 𝑦 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 𝑚-point block method, the interval is divided into series of blocks with each block containing 𝑚-points. The application of 𝑚-point block method generates a block of new solution concurrently. where ]1 and ]2 are not integers. Interpolating (5) at the points 𝑥𝑛 , 𝑥𝑛+1 , 𝑥𝑛+2 and collocating (6) at the points 𝑥𝑛 , 𝑥𝑛+1/2 , 𝑥𝑛+1 , 𝑥𝑛+3/2 , 𝑥𝑛+2 , and 𝑥𝑛+3 lead to a system of nine equations, which can be solved by Mathematica software to obtain the coefficient 𝜙𝑗 . The values of 𝜙𝑗 are substituted into (4) to obtain the continuous multistep method of the form 𝑘 𝑘 2 𝑗=0 𝑗=0 𝑗=1 𝑌 (𝑥) = ∑𝛼𝑗 𝑦𝑛+𝑗 + ℎ3 ( ∑ 𝛽𝑗 𝑓𝑛+𝑗 + ∑𝛽]𝑗 𝑓𝑛+]𝑗 ) , (7) where 𝛼𝑗 , 𝛽𝑗 , and 𝛽]𝑗 are constant coefficients. Hence, the block hybrid collocation method can be derived as follows. Main Method. Consider 𝑦𝑛+3 = 3𝑦𝑛+2 − 3𝑦𝑛+1 + 𝑦𝑛 + ℎ3 90 (8) × (𝑓𝑛 + 36𝑓𝑛+1 + 16𝑓𝑛+3/2 + 36𝑓𝑛+2 + 𝑓𝑛+3 ) . Additional Method. Consider 3 3 1 ℎ3 𝑦𝑛+3/2 = 𝑦𝑛+2 + 𝑦𝑛+1 − 𝑦𝑛 + 8 4 8 92160 2. Derivation of Block Hybrid Collocation Methods × (−19𝑓𝑛 − 648𝑓𝑛+1/2 − 2979𝑓𝑛+1 The hybrid collocation method that produces approximations 󸀠 󸀠󸀠 , and 𝑦𝑛+𝑘 to the general third order ODEs is given 𝑦𝑛+𝑘 , 𝑦𝑛+𝑘 as follows: 𝑘 2 𝑘 2 𝑗=0 𝑗=1 𝑗=0 𝑗=1 − 2104𝑓𝑛+3/2 − 9𝑓𝑛+2 − 𝑓𝑛+3 ) , 1 3 3 ℎ3 𝑦𝑛+1/2 = − 𝑦𝑛+2 + 𝑦𝑛+1 + 𝑦𝑛 + 8 4 8 92160 (9) × (29𝑓𝑛 + 2040𝑓𝑛+1/2 + 3069𝑓𝑛+1 ∑𝛼𝑗 𝑦𝑛+𝑗 + ∑𝛼]𝑗 𝑦𝑛+]𝑗 = ℎ3 ( ∑ 𝛽𝑗 𝑓𝑛+𝑗 + ∑𝛽]𝑗 𝑓𝑛+]𝑗 ) . + 584𝑓𝑛+3/2 + 39𝑓𝑛+2 − 𝑓𝑛+3 ) . (3) In order to obtain (3), we approximate the solution by the interpolating function 𝑌(𝑥) of the form It is noted that the general third order ODEs involve the first and second derivatives. These derivatives can be obtained by imposing that 𝑟+𝑠−1 𝑌 (𝑥) (...truncated)