Efficient Two-Derivative Runge-Kutta-Nyström Methods for Solving General Second-Order Ordinary Differential Equations

Hindawi Discrete Dynamics in Nature and Society Volume 2018, Article ID 2393015, 10 pages https://doi.org/10.1155/2018/2393015 Research Article Efficient Two-Derivative Runge-Kutta-Nyström Methods for Solving General Second-Order Ordinary Differential Equations 𝑦󸀠󸀠(𝑥) = 𝑓(𝑥, 𝑦, 𝑦󸀠) T. S. Mohamed ,1,2 N. Senu ,1,3 Z. B. Ibrahim ,1,3 and N. M. A. Nik Long 1,3 1 Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia Department of Mathematics, Faculty of Science, Misrata University, Misrata, Libya 3 Department of Mathematics, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia 2 Correspondence should be addressed to N. Senu; Received 17 September 2017; Revised 2 January 2018; Accepted 5 February 2018; Published 20 March 2018 Academic Editor: Ciprian G. Gal Copyright © 2018 T. S. Mohamed 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. This paper proposes and investigates a special class of explicit Runge-Kutta-Nyström (RKN) methods for problems in the form 𝑦󸀠󸀠 (𝑥) = 𝑓(𝑥, 𝑦, 𝑦󸀠 ) including third derivatives and denoted as STDRKN. The methods involve one evaluation of second derivative and many evaluations of third derivative per step. In this study, methods with two and three stages of orders four and five, respectively, are presented. The stability property of the methods is discussed. Numerical experiments have clearly shown the accuracy and the efficiency of the new methods. 1. Introduction In this article, we are interested in initial value problems (IVPs) of second-order ordinary differential equations (ODEs): 𝑦󸀠󸀠 (𝑥) = 𝑓 (𝑥, 𝑦 (𝑥) , 𝑦󸀠 (𝑥)) , 𝑦 (𝑥0 ) = 𝛼, (1) 𝑦󸀠 (𝑥0 ) = 𝛽, 𝑥 ∈ [𝑥0 , 𝑥end ] , 𝑁 𝑁 where 𝑦 ∈ R , 𝑓 : R × R × R → R𝑁 are continuous vector valued functions. This type of problems arises naturally in many applied science fields such as the Kepler problems in celestial mechanics, quantum physics, and Newton’s second law in classical mechanics (see Dormand [1], Hairer et al. [2], and Kristensson [3]). Problems (1) in which the first derivative does not appear explicitly are an important subclass of second order (ODEs). Thus, several numerical methods for directly solving this subclass have been presented (see Dormand [1], Hairer et al. [2], Butcher [4], Lambert [5], and Senu [6]). In the case of direct solutions for the general second order (IVPs), some numerical methods have been proposed (see Chen et al. [7], Franco [8], Jator [9], Awoyemi [10], Wu et al. [11], Wu and Wang [12], and Chawla and Sharma [13]). The objective of this paper is to design STDRKN methods with a minimal number of function evaluation. This paper is organized as follows: In Section 3, we construct STDRKN methods; the stability analysis of STDRKN methods is discussed in Section 4; and numerical results are given in Section 5. 2. The Formulation of STDRKN Methods 𝑁 In many problems in applications the third derivative 𝑦󸀠󸀠󸀠 (𝑥) = 𝑔 (𝑥, 𝑦, 𝑦󸀠 ) = 𝑓𝑥 (𝑥, 𝑦, 𝑦󸀠 ) + 𝑓𝑦 (𝑥, 𝑦, 𝑦󸀠 ) 𝑦󸀠 (2) + 𝑓𝑦󸀠 (𝑥, 𝑦, 𝑦󸀠 ) 𝑓 (𝑥, 𝑦, 𝑦󸀠 ) is available and easy to obtain. This derivative can be computed but the evaluation of 𝑔(𝑥, 𝑦, 𝑦󸀠 ) requires the evaluation 2 Discrete Dynamics in Nature and Society of 𝑓(𝑥, 𝑦, 𝑦󸀠 ), 𝑓𝑥 (𝑥, 𝑦, 𝑦󸀠 ), 𝑓𝑦󸀠 (𝑥, 𝑦, 𝑦󸀠 ) 𝑓𝑦 (𝑥, 𝑦, 𝑦󸀠 )𝑦󸀠 . Therefore, in the scalar case (differential systems of dimension one), an evaluation of the third derivative 𝑔(𝑥, 𝑦, 𝑦󸀠 ) can be as expensive as four evaluations of the second derivative 𝑓(𝑥, 𝑦, 𝑦󸀠 ) and at least as two 𝑓-evaluations. An 𝑠-stage twoderivative Runge-Kutta-Nyström (TDRKN) method for (1) is defined by the formula (see Chen et al. [7]) Table 1: Butcher tableau for TDRKN methods. 𝐶 𝐴 𝑏𝑇 𝐶 𝐴 𝑏𝑇 𝑦𝑛+1 = 𝑦𝑛 + ℎ𝑦𝑛󸀠 + ℎ2 ∑𝑏𝑖 𝑓 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑌𝑖 , 𝑌𝑖󸀠 ) 𝑖=1 𝑠 𝑅 𝑑𝑇 𝑌𝑖󸀠 = 𝑦𝑛󸀠 + c𝑖 ℎ𝑓 (𝑥𝑛 , 𝑦𝑛 , 𝑦𝑛󸀠 ) 𝑠 + ℎ ∑𝑑𝑖 𝑔 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑌𝑖 , 𝑌𝑖󸀠 ) , 𝑖=1 𝑅󸀠 𝑑󸀠𝑇 Table 2: Butcher tableau for STDRKN methods. 𝑠 3 𝐴󸀠 𝑏󸀠𝑇 𝑅 𝑑𝑇 𝑠 + ℎ2 ∑ 𝑟𝑖,𝑗 𝑔 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑌𝑗 , 𝑌𝑗󸀠 ) . (3) 𝑗=1 󸀠 𝑦𝑛+1 = 𝑦𝑛󸀠 + ℎ∑𝑏𝑖󸀠 𝑓 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑌𝑖 , 𝑌𝑖󸀠 ) (6) 𝑖=1 𝑠 An alternative expression of formula (5) is given as follows: + ℎ2 ∑𝑑𝑖󸀠 𝑔 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑌𝑖 , 𝑌𝑖󸀠 ) , 𝑖=1 𝑦𝑛+1 = 𝑦𝑛 + ℎ𝑦𝑛󸀠 + where 𝑠 𝑠 ℎ2 𝑓 (𝑥𝑛 , 𝑦𝑛 , 𝑦𝑛󸀠 ) + ℎ3 ∑𝑏𝑖 𝑘𝑖 , 2 𝑖=1 𝑠 󸀠 𝑦𝑛+1 = 𝑦𝑛󸀠 + ℎ𝑓 (𝑥𝑛 , 𝑦𝑛 , 𝑦𝑛󸀠 ) + ℎ2 ∑𝑑𝑖 𝑘𝑖 , 𝑖=1 𝑌𝑖 = 𝑦𝑛 + 𝑐𝑖 ℎ𝑦𝑛󸀠 + ℎ2 ∑ 𝑎𝑖,𝑗 𝑓 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑌𝑗 , 𝑌𝑗󸀠 ) 𝑗=1 𝑠 + ℎ3 ∑𝑟𝑖,𝑗 𝑔 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑌𝑗 , 𝑌𝑗󸀠 ) , (7) where 𝑗=1 𝑠 󸀠 𝑓 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑌𝑗 , 𝑌𝑗󸀠 ) 𝑌𝑖󸀠 = 𝑦𝑛󸀠 + ℎ∑ 𝑎𝑖,𝑗 𝑗=1 (4) 𝑠 𝑗=1 󸀠 󸀠 , 𝑟𝑖,𝑗 , 𝑖, 𝑗 = 1, . . . , 𝑠, are where 𝑐𝑖 , 𝑏𝑖 , 𝑑𝑖 , 𝑏𝑖󸀠 , 𝑑𝑖󸀠 , 𝑎𝑖,𝑗 , 𝑟𝑖,𝑗 , 𝑎𝑖,𝑗 real numbers. This method can also be written in Butcher’s tableau of coefficients as given in Table 1. In this paper, a special part of TDRKN method is studied that has the form 3 ℎ2 𝑓 (𝑥𝑛 , 𝑦𝑛 , 𝑦𝑛󸀠 ) 2 󸀠 𝑦𝑛+1 = 𝑦𝑛󸀠 + ℎ𝑓 (𝑥𝑛 , 𝑦𝑛 , 𝑦𝑛󸀠 ) 𝑠 + ℎ2 ∑𝑑𝑖 𝑔 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑌i , 𝑌𝑖󸀠 ) , 𝑖=1 where 1 𝑌𝑖 = 𝑦𝑛 + 𝑐𝑖 ℎ𝑦𝑛󸀠 + 𝑐𝑖2 ℎ2 𝑓 (𝑥𝑛 , 𝑦𝑛 , 𝑦𝑛󸀠 ) 2 𝑠 + ℎ3 ∑ 𝑎𝑖,𝑗 𝑔 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑌𝑗 , 𝑌𝑗󸀠 ) , 𝑗=1 𝑠 𝑗=1 𝑗=1 (8) This STDRKN method can be written in Butcher’s tableau as shown in Table 2. The STDRKN methods are explicit methods if 𝑎𝑖,𝑗 = 0, 𝑟𝑖,𝑗 = 0 for 𝑖 ≤ 𝑗 and are implicit method if 𝑎𝑖,𝑗 ≠ 0, 𝑟𝑖,𝑗 ≠ 0 for 𝑖 ≤ 𝑗. STDRKN methods involve only one evaluation of 𝑓 and many evaluations of 𝑔 per step. 3. Construction of STDRKN Methods 𝑠 + ℎ ∑𝑏𝑖 𝑔 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑌𝑖 , 𝑌𝑖󸀠 ) , 𝑖=1 𝑠 + ℎ3 ∑ 𝑎𝑖,𝑗 𝑘𝑗 , 𝑦𝑛󸀠 + 𝑐𝑖 ℎ𝑓 (𝑥𝑛 , 𝑦𝑛 , 𝑦𝑛󸀠 ) + ℎ2 ∑𝑟𝑖,𝑗 𝑘𝑗 ) . 󸀠 𝑔 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑌𝑗 , 𝑌𝑗󸀠 ) , + ℎ2 ∑𝑟𝑖,𝑗 𝑦𝑛+1 = 𝑦𝑛 + ℎ𝑦𝑛󸀠 + 1 𝑘𝑖 = 𝑔 (𝑥𝑛 + 𝑐𝑖 , 𝑦𝑛 + 𝑐𝑖 ℎ𝑦𝑛󸀠 + 𝑐𝑖2 ℎ2 𝑓 (𝑥𝑛 , 𝑦𝑛 , 𝑦𝑛󸀠 ) 2 (5) In this section, our effort is to determine the coefficients of the STDRKN methods as given in (7). Hence, using the Taylor series expansion in (3) with the Taylor series expansion of 𝑦𝑛 and 𝑦𝑛󸀠 and by comparing the coefficients of the power of ℎ, we obtained the order conditions of STDRKN methods for 𝑦 and 𝑦󸀠 as in (10)–(18), while the rooted trees for STDRKN methods up to order five based on [7] are given in Table 3. The following simplifying assumption is suggested in practice: 𝑖−1 ∑𝑟𝑖,𝑗 = 𝑗=1 𝑐𝑖2 , 2 𝑖 = 2, . . . , 𝑠. (9) The following are the order conditions for explicit STDRKN. The order conditions for 𝑦: Discrete Dynamics in Nature and Society 3 Table 3: Root trees for STDRKN methods up to order five. Order 𝜌(𝑡) Tree 𝑡 𝛼(𝑡) Density Υ(𝑡) Elementary weight Φ(𝑡) Elementary differential 𝐹(𝑡)(𝑦, 𝑦󸀠 ) 0 1 2 𝜙 1 1 1 1 1 2 𝑒 𝑦 𝑦󸀠 𝑓 3 1 6 𝑐 𝑓𝑦󸀠 𝑦󸀠 3 1 6 𝑐 𝑓𝑦󸀠󸀠 𝑓 4 1 12 𝑐2 󸀠󸀠 𝑓𝑦𝑦 (𝑦󸀠 , 𝑦󸀠 ) 4 2 12 𝑐2 󸀠󸀠 󸀠 𝑓𝑦𝑦 󸀠 (𝑦 , 𝑓) 4 1 12 𝑐2 /2 𝑓𝑦󸀠󸀠󸀠 𝑦󸀠 (𝑓, 𝑓) 4 1 24 𝑐 𝑓𝑦󸀠󸀠 𝑓𝑦󸀠 𝑦󸀠 4 1 24 (1/2) 𝑐2 𝑓𝑦󸀠 𝑦󸀠 4 1 24 𝑐 𝑓𝑦󸀠󸀠 𝑓𝑦󸀠󸀠 𝑓 5 1 20 𝑐3 (3) 𝑓𝑦𝑦𝑦 (𝑦󸀠 , 𝑦󸀠 , 𝑦󸀠 ) 5 3 20 𝑐3 (3) 󸀠 󸀠 𝑓𝑦𝑦𝑦 󸀠 (𝑦 , 𝑦 , 𝑓) 5 3 20 𝑐3 (3) 󸀠 𝑓𝑦𝑦 󸀠 𝑦󸀠 (𝑦 , 𝑓, 𝑓) 5 1 20 (1/2) 𝑐2 𝑓𝑦(3) 󸀠 𝑦󸀠 𝑦󸀠 (𝑓, 𝑓, 𝑓) 5 3 40 𝑅𝑐 󸀠󸀠 (𝑦󸀠 , 𝑓) 𝑓𝑦𝑦 5 1 40 (1/2) 𝑐3 󸀠󸀠 󸀠 󸀠 󸀠 𝑓𝑦𝑦 󸀠 (𝑦 , 𝑓𝑦 𝑦 ) 5 3 (...truncated)