Applied Mathematics and Computation 291 (2016) 39–51 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc An efficient variable step-size rational Falkner-type method for solving the special second-order IVP Higinio Ramos a,b,∗ , Gurjinder Singh c , V. Kanwar c , Saurabh Bhatia c a Scientific Computing Group, Universidad de Salamanca, Plaza de la Merced, Salamanca 37008, Spain b Escuela Politécnica Superior de Zamora, Campus Viriato, Zamora 49022, Spain c University Institute of Engineering and Technology, Panjab University, Chandigarh 160014, India a r t i c l e i n f o Keywords: Ordinary differential equations Initial value problems Rational method Special second order differential equation a b s t r a c t In this paper, firstly a rational one-parameter family of Falkner-type explicit methods is presented for directly solving numerically special second order initial value problems in ordinary differential equations. The proposed family of methods has second algebraic or- der of convergence. Imposing that the principal term of the local truncation error of the proposed family vanishes, we get an expression for the free parameter at the grid point (x n , y n ). By substituting this value of the free parameter in the family, a new rational third order method is obtained. Further, by combining the third order method with any mem- ber of the second order family, their variable step-size formulation as an embedded pair is considered. Some numerical experiments are given to illustrate the performance and efficiency of the proposed methods. © 2016 Elsevier Inc. All rights reserved. 1. Introduction The present paper is concerned with the following so-called special second-order differential equation y ′′ (x) = f (x, y(x)), y(a) = y 0 , y ′ (a) = y ′ 0 , (1) where x ∈ [a, b], y : [a, b] → R and f : [a, b] × R → R are sufficiently differentiable functions. It is possible that (1) can be integrated by reformulating it as a system of two first order ODEs and applying one of the methods available for those systems (see for example [7,13,14,17]). Nevertheless, it seems more natural to provide numerical methods to integrate (1) di- rectly without transforming it in a first order system. These approaches result to be more efficient. For instance, it is well- known that a Runge–Kutta–Nyström method for solving (1) has a real improvement as compared to standard Runge–Kutta methods. In case of a linear k-step method for first order ODEs, it becomes a 2k-step method for (1), thus increasing the computational work. For the general second order initial value problem of the form y ′′ (x) = f (x, y(x), y ′ (x)), y(a) = y 0 , y ′ (a) = y ′ 0 one of the methods for solving it directly is the explicit method due to Falkner [1] which can be written in the form ∗ Corresponding author at: Scientific Computing Group, Universidad de Salamanca, Plaza de la Merced, 37008 Salamanca, Spain. Tel.: +34 980545000. E-mail addresses: higra@usal.es (H. Ramos), gurjinder11@gmail.com (G. Singh), vmithil@yahoo.co.in (V. Kanwar), s_bhatia@pu.ac.in (S. Bhatia). http://dx.doi.org/10.1016/j.amc.2016.06.033 0096-3003/© 2016 Elsevier Inc. All rights reserved.