Two Embedded Pairs of Runge-Kutta Type Methods for Direct Solution of Special Fourth-Order Ordinary Differential Equations

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 196595, 12 pages http://dx.doi.org/10.1155/2015/196595 Research Article Two Embedded Pairs of Runge-Kutta Type Methods for Direct Solution of Special Fourth-Order Ordinary Differential Equations Kasim Hussain,1,2 Fudziah Ismail,1,3 and Norazak Senu1,3 1 Department of Mathematics, Faculty of Science, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia 2 Department of Mathematics, College of Science, Al-Mustansiriyah University, Baghdad, Iraq 3 Institute for Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia Correspondence should be addressed to Fudziah Ismail; fudziah Received 14 August 2015; Revised 3 November 2015; Accepted 8 November 2015 Academic Editor: Tarek Ahmed-Ali Copyright © 2015 Kasim Hussain 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. We present two pairs of embedded Runge-Kutta type methods for direct solution of fourth-order ordinary differential equations (ODEs) of the form 𝑦(𝑖V) = 𝑓(𝑥, 𝑦) denoted as RKFD methods. The first pair, which we will call RKFD5(4), has orders 5 and 4, and the second one has orders 6 and 5 and we will call it RKFD6(5). The techniques used in the derivation of the methods are that the higher order methods are very precise and the lower order methods give the best error estimate. Based on these pairs, we have developed variable step codes and we have used them to solve a set of special fourth-order problems. Numerical results show the robustness and the efficiency of the new RKFD pairs as compared with the well-known embedded Runge-Kutta pairs in the scientific literature after reducing the problems into a system of first-order ordinary differential equations (ODEs) and solving them. 1. Introduction This paper deals with embedded RKFD methods for directly solving special fourth-order ordinary differential equations (ODEs) of the form 𝑦(𝑖V) (𝑥) = 𝑓 (𝑥, 𝑦) , 𝑥 ≥ 𝑥0 (1) with initial conditions 𝑦 (𝑥0 ) = 𝑦0 , 𝑦󸀠 (𝑥0 ) = 𝑦0󸀠 , 𝑦󸀠󸀠 (𝑥0 ) = 𝑦0󸀠󸀠 , 𝑦󸀠󸀠󸀠 (𝑥0 ) = 𝑦0󸀠󸀠󸀠 , (2) in which the first, second, and third derivatives do not appear explicitly. This type of problems can be found in various fields of applied science and engineering such as beam theory [1, 2], fluid dynamics [3], neural networks [4], and electric circuits [5]. Traditionally, the fourth-order ordinary differential equations are transformed to a first-order system of ordinary differential equations, so that standard numerical methods can be applied (see [6–11]). However, several researchers (see [1, 12, 13]) observed the drawback of this technique as it wastes a lot of computing time and human effort. Therefore, direct integration methods have attracted significant attention from several authors for solving higher order ODEs, because these direct methods demonstrated the features in accuracy and speed (see [14–23]). However, all the methods discussed above are multistep methods in nature. This paper primarily aims to construct a one-step method to solve special fourthorder ODEs directly; this new method is self-starting in nature. 2 Mathematical Problems in Engineering The general form of RKFD method with 𝑠-stage for solving special fourth-order ODEs (1) can be expressed as follows [24]: 𝑦𝑛+1 = 𝑦𝑛 + ℎ𝑦𝑛󸀠 + 𝑠 ℎ2 󸀠󸀠 ℎ3 󸀠󸀠󸀠 𝑦𝑛 + 𝑦𝑛 + ℎ4 ∑𝑏𝑖 𝑘𝑖 , 2 6 𝑖=1 𝑠 ℎ2 󸀠 = 𝑦𝑛󸀠 + ℎ𝑦𝑛󸀠󸀠 + 𝑦𝑛󸀠󸀠󸀠 + ℎ3 ∑𝑏𝑖󸀠 𝑘𝑖 , 𝑦𝑛+1 2 𝑖=1 𝑠 󸀠󸀠 𝑦𝑛+1 = 𝑦𝑛󸀠󸀠 + ℎ𝑦𝑛󸀠󸀠󸀠 + ℎ2 ∑𝑏𝑖󸀠󸀠 𝑘𝑖 , 𝑖=1 pairs of RKFD methods that provide a cheap error estimation for variable step size codes. They depend on the methods (𝑐, 𝐴, 𝑏, 𝑏󸀠 , 𝑏󸀠󸀠 , 𝑏󸀠󸀠󸀠 ) of order 𝑟 and (𝑐, 𝐴, ̂𝑏, ̂𝑏󸀠 , ̂𝑏󸀠󸀠 , ̂𝑏󸀠󸀠󸀠 ) of order V. Butcher tableau of embedded RKFD pair can be written as follows: 𝑐 𝐴 𝑏𝑇 (3) 𝑏󸀠𝑇 𝑏󸀠󸀠𝑇 𝑠 𝑏󸀠󸀠󸀠𝑇 ̂𝑏𝑇 󸀠󸀠󸀠 𝑦𝑛+1 = 𝑦𝑛󸀠󸀠󸀠 + ℎ∑𝑏𝑖󸀠󸀠󸀠 𝑘𝑖 , 𝑖=1 (5) ̂𝑏󸀠𝑇 where 𝑘1 = 𝑓 (𝑥𝑛 , 𝑦𝑛 ) , 𝑘𝑖 = 𝑓 (𝑥𝑛 + 𝑐𝑖 ℎ, 𝑦𝑛 + ℎ𝑐𝑖 𝑦𝑛󸀠 + ̂𝑏󸀠󸀠𝑇 ℎ2 2 󸀠󸀠 ℎ3 3 󸀠󸀠󸀠 𝑐 𝑦 + 𝑐𝑖 𝑦𝑛 2 𝑖 𝑛 6 ̂𝑏󸀠󸀠󸀠𝑇 (4) 𝑠 + ℎ4 ∑𝑎𝑖𝑗 𝑘𝑗 ) , 𝑗=1 for 𝑖 = 2, 3, . . . , 𝑠. The parameters 𝑏𝑖 , 𝑏𝑖󸀠 , 𝑏𝑖󸀠󸀠 , 𝑏𝑖󸀠󸀠󸀠 , 𝑎𝑖𝑗 , and 𝑐𝑖 of the RKFD method are to be determined for 𝑖 = 1, 2, . . . , 𝑠 and 𝑗 = 1, 2, . . . , 𝑠 and supposed to be real. The RKFD method is an explicit method if 𝑎𝑖𝑗 = 0 for 𝑖 ≤ 𝑗 and is an implicit method if 𝑎𝑖𝑗 ≠ 0 for some 𝑖 such that 𝑖 ≤ 𝑗. To determine the parameters of the RKFD method given in (3)-(4), the RKFD method expression is expanded using the Taylor series expansion. After doing some algebraic simplifications, this expansion is equated to the true solution that is given by the Taylor series expansion. The direct expansion of the truncation error is used to derive the order conditions for the RKFD method [25]. A good deal of algebraic and numerical calculations are required for the above operation which were carried out using algebra package Maple [26]. Algebraic order conditions for the RKFD method can be obtained from the direct expansion of the local truncation error. In this paper we will derive embedded Runge-Kutta pairs for direct integration of special fourth-order ODEs. Embedded pairs of RK type methods have a built-in local truncation error estimate; as a result, the step size can be controlled at virtually no extra cost, and hence an efficient variable step size code can be developed. In recent years, the construction of embedded RungeKutta method is an effective research area yielding continuous development to the existing codes. The present paper is primarily dedicated as an extra work in this research area. This technique involves two Runge-Kutta formulae of orders 𝑟 and V (𝑟 > V, usually 𝑟 = V + 1) (see [25, 27–33]). We are interested in deriving the effective embedded 𝑟(V) 󸀠 󸀠󸀠 󸀠󸀠󸀠 The method will compute 𝑦𝑛+1 , 𝑦𝑛+1 , 𝑦𝑛+1 , and 𝑦𝑛+1 to approximate 𝑦(𝑥𝑛+1 ), 𝑦󸀠 (𝑥𝑛+1 ), 𝑦󸀠󸀠 (𝑥𝑛+1 ), and 𝑦󸀠󸀠󸀠 (𝑥𝑛+1 ), where 𝑦𝑛+1 is the computed solution and 𝑦(𝑥𝑛+1 ) is the exact solution. The remainder of this paper is organized as follows. In Section 2, we present the order conditions of RKFD method as well as the basic concepts and notations which are used for embedded method. In Section 3, we present the construction of the new embedded RKFD pairs of orders 5(4) and 6(5), respectively. In Section 4, we carry out the numerical experiments to show the efficiency of the new embedded RKFD pairs when compared with the well-known Runge-Kutta pairs from the scientific literature. Conclusions of the paper are given in Section 5. 2. The Order Conditions of RKFD Method The order conditions of RKFD method up to fifth order have been derived using Taylor series expansion by Hussain et al. [24]. We use the same approach to derive the order conditions up to seventh order and for convenience we will present the algebraic order conditions of the RKFD method given in [24] together with the seventh order method in this paper. Next, we giv (...truncated)