Implicit One-Step Block Hybrid Third-Derivative Method for the Direct Solution of Initial Value Problems of Second-Order Ordinary Differential Equations

Hindawi Journal of Applied Mathematics Volume 2017, Article ID 8510948, 8 pages https://doi.org/10.1155/2017/8510948 Research Article Implicit One-Step Block Hybrid Third-Derivative Method for the Direct Solution of Initial Value Problems of Second-Order Ordinary Differential Equations Mohammad Alkasassbeh and Zurni Omar Department of Mathematics, School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia, Sintok, Malaysia Correspondence should be addressed to Mohammad Alkasassbeh; Received 25 August 2016; Revised 20 October 2016; Accepted 23 October 2016; Published 18 January 2017 Academic Editor: Mehmet Sezer Copyright © 2017 Mohammad Alkasassbeh and Zurni Omar. 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. A new one-step block method with generalized three hybrid points for solving initial value problems of second-order ordinary differential equations directly is proposed. In deriving this method, a power series approximate function is interpolated at {𝑥𝑛 , 𝑥𝑛+𝑟 } while its second and third derivatives are collocated at all points {𝑥𝑛 , 𝑥𝑛+𝑟 , 𝑥𝑛+𝑠 , 𝑥𝑛+𝑡 , 𝑥𝑛+1 } in the given interval. The proposed method is then tested on initial value problems of second-order ordinary differential equations solved by other methods previously. The numerical results confirm the superiority of the new method to the existing methods in terms of accuracy. 1. Introduction Numerous problems such as chemical kinetics, orbital dynamics, circuit and control theory, and Newton’s second law applications involve second-order ordinary differential equations (ODEs). Normally, those equations have no analytical solutions. To approximate the solution of such problems several numerical methods were developed on the hands of many scholars such as [1–3]. Block methods for solving ODEs were first proposed by Milne ([4]). Later [5] adopted Milne’s methods to provide starting values for predictor-corrector scheme. However, the block methods have some drawbacks and this led to the introduction of hybrid methods. According to [6], hybrid methods were initially introduced to overcome zero-stability barrier that occurred in block methods in Dahlquists ([7]). Besides the ability to change step size, the other benefit of these methods is utilizing data off-step points which contribute to the accuracy of the methods. To increase the accuracy of the numerical methods further, researchers such as [8, 9] proposed high method derivative to overcome stiffness in ODEs. The former presented another type of hybrid methods called secondderivative methods, while the later proposed a Simpson’s type second-derivative method for the solution of a stiff system of first-order IVPs. These scholars motivated us to develop a new generalized three-hybrid one-step third-derivative implicit method for solving second-order ODEs directly using the approach of interpolation and collocation for the general use to improve the efficiency of the approximate solution. This article is organized as follows: in the coming section we demonstrate the derivation of the method, where we consider three off-step points through the approach of interpolation and collocation. The details of the analysis of the method are discussed in Section 3 which include zero stability, order, consistency, and convergence. In Section 4 some numerical problems are solved and the performance of the developed method is compared with other methods mentioned in literature. Finally, the conclusion is discussed in Section 5. 2. Development of the Method An approximate power series basis function taking the form 2V+𝑢−1 𝑝 (𝑥) = ∑ 𝑎𝑗 ( 𝑗=0 𝑥 − 𝑥𝑛 𝑗 ) , ℎ (1) 2 Journal of Applied Mathematics where 𝑢 = 2 and V = 5 are the number of interpolation and collocation points, respectively, is considered to be a solution to the following ODE: 𝑦󸀠󸀠 = 𝑓 (𝑥, 𝑦, 𝑦󸀠 ) , 𝑦 (𝑎) = 𝑝0 , (2) 𝑦0󸀠 (𝑎) = 𝑝0󸀠 , 𝑥 ∈ [𝑎, 𝑏] . On derivation of (1) twice and thrice we obtain 2V+𝑢−1 𝑝󸀠󸀠 (𝑥) = ∑ 𝑗=2 𝑎𝑗 𝑗! 𝑥 − 𝑥𝑛 ( ) 2 ℎ ℎ (𝑗 − 2)! 𝑗−2 = 𝑓 (𝑥, 𝑦, 𝑦󸀠 ) , 2V+𝑢−1 𝑎𝑗 𝑗! 𝑥 − 𝑥𝑛 𝑗−3 𝑝 (𝑥) = ∑ 3 ( ) ℎ 𝑗=3 ℎ (𝑗 − 3)! 󸀠󸀠󸀠 (3) = 𝑔 (𝑥, 𝑦, 𝑦󸀠 ) . ̂ ℎ, 𝑢 ̂ = {0, 𝑟} and collocating Interpolating (1) at 𝑥𝑛+̂𝑢 = 𝑥𝑛 + 𝑢 (3) at all points 𝑥𝑛+̂V = 𝑥𝑛 +̂Vℎ, ̂V = {0, 𝑟, 𝑠, 𝑡, 1}, where {𝑟, 𝑠, 𝑡} ∈ (0, 1), a system of equations in matrix form is produced as below: 𝐴𝑋 = 𝑈, where 𝑎0 [ ] [ 𝑎1 ] [ ] [ ] [ 𝑎2 ] [ ] [𝑎 ] [ 3] [ ] [𝑎 ] [ 4] [ ] [ 𝑎5 ] [ ] 𝐴 = [ ], [ 𝑎6 ] [ ] [ ] [ 𝑎7 ] [ ] [ ] [ 𝑎8 ] [ ] [𝑎 ] [ 9] [ ] [𝑎 ] [ 10 ] [𝑎11 ] (4) 𝑋 1 0 [ [1 𝑟 [ [ [ [0 0 [ [ [ [0 0 [ [ [ [ [0 0 [ [ [ [ [0 0 [ [ [ =[ [0 0 [ [ [ [0 0 [ [ [ [0 0 [ [ [ [ [0 0 [ [ [ [ [0 0 [ [ [ [ 0 0 [ 0 0 0 2 3 4 ⋅⋅⋅ 0 ] ] 𝑟 𝑟 ⋅⋅⋅ 𝑟 𝑟 ] ] 2! ] ] 0 0 ⋅ ⋅ ⋅ 0 ] 0!ℎ2 2 (2V+𝑢−3) ] ] 3!𝑟 4!𝑟 2! (2V + 𝑢 − 1)!𝑟 ] ⋅⋅⋅ 0!ℎ2 1!ℎ2 2!ℎ2 (2V + 𝑢 − 3)!ℎ2 ] ] ] 3!𝑠 4!𝑠2 2! (2V + 𝑢 − 1)!𝑠(2V+𝑢−3) ] ] ⋅ ⋅ ⋅ 0!ℎ2 1!ℎ2 2!ℎ2 (2V + 𝑢 − 3)!ℎ2 ] ] 2 (2V+𝑢−3) ] ] 3!𝑡 4!𝑡 2! (2V + 𝑢 − 1)!𝑡 ] ⋅⋅⋅ ] 2 2 2 2 0!ℎ 1!ℎ 2!ℎ (2V + 𝑢 − 3)!ℎ ] ] 3! 4! 2! (2V + 𝑢 − 1)! ], ⋅ ⋅ ⋅ ] 0!ℎ2 1!ℎ2 2!ℎ2 (2V + 𝑢 − 3)!ℎ2 ] ] ] 3! ] 0 0 ⋅ ⋅ ⋅ 0 ] 0!ℎ3 ] 2 (2V+𝑢−4) ] 3!𝑟 4!𝑟 (2V + 𝑢 − 1)!𝑟 ] ⋅⋅⋅ 0 ] 0!ℎ3 1!ℎ3 (2V + 𝑢 − 4)!ℎ3 ] ] 3!𝑠 4!𝑠2 (2V + 𝑢 − 1)!𝑠(2V+𝑢−4) ] ] 0 ⋅⋅⋅ ] 0!ℎ3 1!ℎ3 (2V + 𝑢 − 4)!ℎ3 ] ] 3!𝑡 4!𝑡2 (2V + 𝑢 − 1)!𝑡(2V+𝑢−4) ] ] 0 ⋅⋅⋅ ] 0!ℎ3 1!ℎ3 (2V + 𝑢 − 4)!ℎ3 ] ] ] 4! 3! (2V + 𝑢 − 1)! 0 ⋅⋅⋅ 3 3 3 0!ℎ 1!ℎ (2V + 𝑢 − 4)!ℎ ] (2V+𝑢−1) 𝑦𝑛 ] [ [𝑦𝑛+𝑟 ] ] [ ] [ [ 𝑓𝑛 ] ] [ [𝑓 ] [ 𝑛+𝑟 ] ] [ [𝑓 ] [ 𝑛+𝑠 ] ] [ [ 𝑓𝑛+𝑡 ] ] [ 𝑈=[ ]. [𝑓𝑛+1 ] ] [ ] [ [ 𝑔𝑛 ] ] [ ] [ [𝑔𝑛+𝑟 ] ] [ [𝑔 ] [ 𝑛+𝑠 ] ] [ [𝑔 ] [ 𝑛+𝑡 ] [𝑔𝑛+1 ] (5) Using matrix manipulation to solve (4) for the unknown coefficients 𝑎𝑗󸀠 s and then substituting them back into (1) yield 1 𝑝 (𝑥) = ∑ 𝛼𝑖 𝑦𝑛+𝑖 + ℎ2 [ ∑ 𝛽𝑖 𝑓𝑛+𝑖 + ∑𝛽𝑖 𝑓𝑛+𝑖 ] 𝑖=0,𝑟 𝑖=𝑟,𝑠,𝑡 1 𝑖=0 (6) + ℎ3 [ ∑ 𝛾𝑖 𝑔𝑛+𝑖 + ∑𝛾𝑖 𝑔𝑛+𝑖 ] , 𝑖=𝑟,𝑠,𝑡 𝑖=0 where 𝑛 = 0, 1, 2, . . . , 𝑁 − 1, ℎ = 𝑥𝑛 − 𝑥𝑛−1 is the constant step size for the partition 𝜋𝑁 of the interval [𝑎, 𝑏] which Journal of Applied Mathematics 3 is given by 𝜋𝑁 = [𝑎 = 𝑥0 < 𝑥1 < ⋅ ⋅ ⋅ < 𝑥𝑁−1 < 𝑥𝑁 = 𝑏], 𝛼𝑖 , 𝛽𝑖 , and 𝛾𝑖 are undetermined constants listed in Appendix I in Supplementary Material available online at https://doi.org/10.1155/2017/8510948, 𝑓𝑛+𝑖 = 𝑓(𝑥 + 𝑖ℎ), and 󸀠 )/𝑑𝑥 whose first partial derivative is 𝑔𝑛+𝑖 = 𝑑𝑓(𝑥𝑛+𝑖 , 𝑦𝑛+𝑖 , 𝑦𝑛+𝑖 𝑝󸀠 (𝑥) = 1 1 ∑ 𝛼𝑖󸀠 𝑦𝑛+𝑖 + ℎ [ ∑ 𝛽𝑖󸀠 𝑓𝑛+𝑖 + ∑𝛽𝑖󸀠 𝑓𝑛+𝑖 ] ℎ 𝑖=0,𝑟 𝑖=𝑟,𝑠,𝑡 𝑖=0 (7) 1 + ℎ2 [ ∑ 𝛾𝑖󸀠 𝑔𝑛+𝑖 + ∑𝛾𝑖󸀠 𝑔𝑛+𝑖 ] . 𝑖=𝑟,𝑠,𝑡 𝑖=0 Evaluating (6) at the noninterpolating points {𝑥𝑛+𝑠 , 𝑥𝑛+𝑡 , 𝑥𝑛+1 } and (7) at all points 𝑥𝑛+𝑖 , 𝑖 = {0, 𝑟, 𝑠, 𝑡, 1}, produces the following general equations in block form: 1 1 𝑖=0 𝑖=0 [𝑖] 𝐴[0] 𝑌𝑚[1] = 𝐴[1] 𝑌𝑚[0] + ∑𝐵[𝑖] 𝐹𝑚[𝑖] + ∑𝐷[𝑖] 𝐺𝑚 , where 𝐴[0] is an 8 × 8 identity matrix and (8) [1] [1] [1] [1] 𝐵12 𝐵13 𝐵14 𝐵11 ] [ [𝐵[1] 𝐵[1] 𝐵[1] 𝐵[1] ] [ 21 22 23 24 ] ] [ [ [1] [1] [1] [1] ] [𝐵31 𝐵32 𝐵33 𝐵34 ] ] [ ] [ [ [1] [1] [1] [1] ] [𝐵41 𝐵4 (...truncated)