Volume 17, Issue 4, pp 527-534
Publication Date: 2017-10-28
- School of Quantitative Sciences, Universiti Utara Malaysia, Sintok, Kedah, Malaysia
Block methods for the numerical solution of ordinary differential equations (ODEs) are quite prominent in recent literature and second order initial value problems (IVPs) which falls in the family of ODEs is also a well explored area for the application of block methods. The introduction of hybrid block method methods for the solution of second order IVPs has gained good grounds in literature as the presence of off-grid points in the block method has increased the accuracy of the hybrid block methods. However, recent studies still continue to introduce new block methods that will perform more favourably than previously existing when compared in terms of error. Hence, a hybrid block method of order six is presented in this article to compete with previously existing methods of the same order and higher order. The methodology adopted in this article presents a new approach for developing the hybrid block method which is simple to implement and less computationally tiresome. The numerical results show this new \(4\)-step \(5\)-point hybrid block method performing better than previously existing methods.
Hybrid, block method, order six, second order, initial value problems
 A. O. Adesanya, Block methods for direct solutions of general higher order initial value problems of ordinary differential equations, PhD Thesis, Federal University of Technology, Akure, Nigeria, (2011).
 A. O. Adesanya, M. R. Odekunle, A. O. Adeyeye, Continuous block hybrid predictor corrector method for the solution of y"=f(x,y,y'), Int. J. Math. Soft Comput., 2 (2012), 35–42.
 A. O. Adesanya, D. M. Udoh, A. M. Ajileye, A new hybrid block method for the solution of general third order initial value problems of ordinary differential equations, Int. J. Pure Appl. Math., 86 (2013), 365–375.
 T. A. Anake, D. O. Awoyemi, A. O. Adesanya, One-step implicit hybrid block method for the direct solution of general second order ordinary differential equations, IAENG Int. J. Appl. Math., 42 (2012), 224–228.
 A. M. Badmus, A new eighth order implicit block algorithms for the direct solution of second order ordinary differential equations, Amer. J. Comput. Math., 4 (2014), 376–386.
 A. A. James, O. A. Adesanya, K. M. Fasasi, Starting order seven method accurately for the solution of first initial value problems of first order ordinary differential equations, Progress Appl. Math., 6 (2013), 30–39.
 S. J. Kayode, O. Adeyeye, A 3-step hybrid method for direct solution of second order initial value problems, Aust. J. Basic Appl. Sci., 5 (2011), 2121–2126.
 S. J. Kayode, O. Adeyeye, Two-step two-point hybrid methods for general second order differential equations, Afr. J. Math. Comput. Sci. Res., 6 (2013), 191–196.
 J. O. Kuboye, Z. Omar, Derivation of a six-step block method for direct solution of second order ordinary differential equations, Math. Comput. Appl., 20 (2015), 151–159.
 Z. A. Majid, N. Z. Mokhtar, M. Suleiman, Direct two-point block one-step method for solving general second-order ordinary differential equations, Math. Probl. Eng., 2012 (2012), 16 pages.
 R. I. Okuonghae, M. N. O. Ikhile, A class of hybrid linear multistep methods with \(A(\alpha)\)-stability properties for stiff IVPs in ODEs, J. Numer. Math., 21 (2013), 157–172.
 Z. Omar, J. O. Kuboye, A new implicit block method for solving second order ordinary differential equations directly, G.U. Journal of Science, 28 (2015), 689–694.
 P. S. Phang, Z. A. Majid, M. Suleiman, Solving nonlinear two point boundary value problem using two step direct method, J. Qual. Measure. Anal., 7 (2011), 129–140.
 K. Rauf, S. A. Aniki, S. Ibrahim, J. O. Omolehin, A zero-stable block method for the solution of third order ordinary differential equations, Pac. J. Sci. Tech., 16 (2015), 91–103.