Research Article An Accurate Block Hybrid Collocation Method for Third Order Ordinary Differential Equations Lee Ken Yap, 1,2 Fudziah Ismail, 2 and Norazak Senu 2 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, 43400 Serdang, Selangor, Malaysia Correspondence should be addressed to Lee Ken Yap; lkyap@utar.edu.my 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. his 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. he block hybrid collocation method with two of-step points is proposed for the direct solution of general third order ordinary diferential equations. Both the main and additional methods are derived via interpolation and collocation of the basic polynomial. hese methods are applied in block form to provide the approximation at ive points concurrently. he stability properties of the block method are investigated. Some numerical examples are tested to illustrate the eiciency of the method. he block hybrid collocation method is also implemented to solve the nonlinear Genesio equation and the problem in thin ilm low. 1. Introduction Consider the general third order ordinary diferential equa- tions (ODEs):   =(,,  ,  ), (1) with the initial conditions  () =  0 ,   () =   0 ,   () =   0 ,  ∈ [, ] . (2) In particular, the third order diferential equations arise in many physical problems such as electromagnetic waves, thin ilm low, and gravity-driven lows (see [1–6]). herefore, third order ODEs have attracted considerable attention. Many theoretical and numerical studies dealing with such equations have appeared in the literature. he popular approach for solving third order ODEs is by converting the problems to a system of irst 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. he 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. he 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 afect 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 of-step point,  +3/2 . Both schemes are implemented in predictor-corrector mode to obtain the approximation at  +3 . he 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] devel- oped the shited Jacobi-Gauss collocation spectral method for general nonlinear third order diferential equations. Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 549597, 9 pages http://dx.doi.org/10.1155/2014/549597