Zero-dissipative semi-implicit hybrid method for solving oscillatory or periodic problems Y.D. Jikantoro a,⇑ , F. Ismail b , N. Senu b a Department of Mathematics, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia b Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia article info Keywords: Oscillatory solution Dispersion Dissipation Hybrid method Initial value problem Oscillatory problem abstract In this paper, a new semi-implicit two-step hybrid method with fifth algebraic order is derived for the integration of second-order oscillatory initial value problems. The new method possesses dispersion of order eight and dissipation of order infinity. Numerical experiment reveals the superiority of the new method for solving oscillatory or periodic problems over several methods of the same algebraic order in the literature. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction In the last few decades, there has been growing interest in the research of new numerical techniques for approximating the solution of second order initial value problem y 00 ðxÞ¼ f ðx; yÞ; yðx 0 Þ¼ y 0 ; y 0 ðx 0 Þ¼ y 0 0 ; ð1Þ which is independent on y 0 explicitly. This type of problem arises in different fields of science and engineering, which includes quantum mechanics, celestial mechanics, molecular dynamics, quantum chemistry, astrophysics, electronics, semi-discretizations of wave equation, and so on. Due to their importance, many numerical methods have been derived for approximating their solutions, some of which are Runge–Kutta methods, Runge–Kutta Nyström, linear multi step methods and so on. For the Runge–Kutta methods and other related methods specifically derived for approximating the solutions of first order IVPs, the second order IVPs need to be transformed into a system of first order IVPs so that the methods can be applied. In the quest for methods that best approximate the solutions of (1) many authors considered different modifications on Runge–Kutta methods, multistep methods and Runge–Kutta Nyström methods, [2–4,7–13,15,16,18–20]. Hybrid type methods related to multistep methods are proposed by many authors for approximating the solutions of (1), for example, see [21,23,22,5]. But most of the multi step hybrid methods are characterized by off-step points and higher stages, which make them expensive. Hence, Franco [10] identified the drawback of most of the multistep hybrid methods as regards to high computation cost, and therefore proposed explicit two-step hybrid methods up to algebraic order six with less computation cost by using the algebraic order conditions of two-step hybrid methods developed in [6]. In furtherance to this, Ahmad et al. [1] proposed semi-implicit hybrid methods up to algebraic order five. In this paper, we derive a semi-implicit hybrid method with improved numerical properties. The method is generally given by http://dx.doi.org/10.1016/j.amc.2014.12.020 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail address: jdauday@yahoo.ca (Y.D. Jikantoro). Applied Mathematics and Computation 252 (2015) 388–396 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc