A Family of Trigonometrically Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value Problems

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2015, Article ID 343295, 17 pages http://dx.doi.org/10.1155/2015/343295 Research Article A Family of Trigonometrically Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value Problems F. F. Ngwane1 and S. N. Jator2 1 Department of Mathematics, USC Salkehatchie, Walterboro, SC 29488, USA Department of Mathematics and Statistics, Austin Peay State University, Clarksville, TN 37044, USA 2 Correspondence should be addressed to F. F. Ngwane; Received 23 January 2015; Accepted 23 April 2015 Academic Editor: Mehmet Sezer Copyright © 2015 F. F. Ngwane and S. N. Jator. 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 family of Enright’s second derivative formulas with trigonometric basis functions is derived using multistep collocation method. The continuous schemes obtained are used to generate complementary methods. The stability properties of the methods are discussed. The methods which can be applied in predictor-corrector form are implemented in block form as simultaneous numerical integrators over nonoverlapping intervals. Numerical results obtained using the proposed block form reveal that the new methods are efficient and highly competitive with existing methods in the literature. 1. Introduction Many real life processes in areas such as chemical kinetics, biological sciences, circuit theory, economics, and reactions in physical systems can be transformed into systems of ordinary differential equations (ODE) which are generally formulated as initial value problems (IVPs). Some classes of IVPs are stiff and/or highly oscillatory as described by the following model problem: 𝑦󸀠 = 𝐴𝑦, 𝑦 (𝑎) = 𝑦0 , (1) 𝑥 ∈ [𝑎, 𝑏] , 𝑚 where 𝑦(𝑥) ∈ R and 𝐴 is 𝑚 × 𝑚 real matrix with at least one eigenvalue with a very negative real part and/or very large imaginary part, respectively (see Fatunla [1]). Many conventional methods cannot solve these types of problems effectively. Stiff systems have been solved by several authors including Lambert [2, 3], Gear [4, 5], Hairer [6], and Hairer and Wanner [7]. Different methods including the Backward Differentiation Formula (BDF) have been used to solve stiff systems. Second derivative methods with polynomial basis functions were proposed to overcome the Dahlquist [8] barrier theorem whereby the conventional linear multistep method was modified by incorporating the second derivative term in the derivation process in order to increase the order of the method, while preserving good stability properties (see Gear [9], Gragg and Stetter [10], and Butcher [11]). Many classical numerical methods including RungeKutta methods, higher derivative multistep schemes, and block methods have been constructed for solving oscillatory initial value problems (see Butcher [11, 12], Brugnano and Trigiante [13, 14], Ozawa [15], Nguyen et al. [16], Berghe and van Daele [17], Vigo-Aguiar and Ramos [18], and Calvo et al. [19]). Many methods for solving oscillatory IVPs require knowledge of the system under consideration in advance. Obrechkoff [20] proposed a general multiderivative method for solving systems of ordinary differential equations. Special cases of Obrechkoff method have been developed by many others including Cash [21] and Enright [22]. The methods by Enright [22] have order 𝑝 = 𝑘 + 2 for a 𝑘 step method. In this paper, we propose a numerical integration formula which more effectively copes with stiff and/or oscillatory 2 Journal of Applied Mathematics IVPs. We will construct a continuous form of the second derivative multistep method (CSDMM) using a multistep collocation technique such that Enright’s second derivative methods (ESDM) will be recovered from the derived continuous methods. The aim of this paper is to derive a family of Enright’s second derivative formulas with trigonometric basis functions using multistep collocation method. Many methods for solving IVPs are implemented in a step-by-step fashion in which, on the partition 𝜋𝑁, an approximation is obtained at 𝑥𝑛+1 only after an approximation at 𝑥𝑛 has been computed, where 𝜋𝑁 : 𝑎 = 𝑥0 < 𝑥1 < ⋅ ⋅ ⋅ < 𝑥𝑛 < 𝑥𝑛+1 < ⋅ ⋅ ⋅ < 𝑥𝑁 = 𝑏, 𝑥𝑛+1 = 𝑥𝑛 + ℎ, 𝑛 = 1, . . . , 𝑁, ℎ = (𝑏 − 𝑎)/𝑁, ℎ is the step size, 𝑁 is a positive integer, and 𝑛 is the grid index. We implement ESDM in block form. In Section 2, we present a derivation of the family of Enright methods. Error analysis and stability are discussed in Section 3. The implementation of the ESDM and numerical examples to show the accuracy and efficiency of the ESDM are given in Section 4. Finally, we conclude in Section 5. 𝑇 𝑉 = (𝑦𝑛+𝑘−1 , 𝑓𝑛 , 𝑓𝑛+1 , 𝑓𝑛+2 , . . . , 𝑓𝑛+𝑘 , 𝑔𝑛+𝑘 ) , 𝑇 𝑃 (𝑥) = (𝑃1 (𝑥) , 𝑃2 (𝑥) , . . . , 𝑃𝑘+3 (𝑥)) , 𝑊(𝑘+3,𝑘+3) 𝑃1 (𝑥𝑛+𝑘−1 ) 𝑃2 (𝑥𝑛+𝑘−1 ) ⋅ ⋅ ⋅ 𝑃𝑘+3 (𝑥𝑛+𝑘−1 ) 𝑃1󸀠 (𝑥𝑛 ) ( 󸀠 ( 𝑃 (𝑥𝑛+1 ) ( 1 ( 󸀠 ( = ( 𝑃1 (𝑥𝑛+2 ) ( .. ( ( . ( 𝑃1󸀠 (𝑥𝑛+𝑘 ) 󸀠󸀠 ( 𝑃1 (𝑥𝑛+𝑘 ) 𝑃2󸀠 (𝑥𝑛 ) ⋅⋅⋅ 󸀠 𝑃𝑘+3 (𝑥𝑛 ) ) (4) 󸀠 𝑃2󸀠 (𝑥𝑛+1 ) ⋅ ⋅ ⋅ 𝑃𝑘+3 (𝑥𝑛+1 ) ) ) ) 󸀠 󸀠 𝑃2 (𝑥𝑛+2 ) ⋅ ⋅ ⋅ 𝑃𝑘+3 (𝑥𝑛+2 ) ) ), ) .. .. ) ) . d . ) 󸀠 󸀠 𝑃2 (𝑥𝑛+𝑘 ) ⋅ ⋅ ⋅ 𝑃𝑘+3 (𝑥𝑛+𝑘 ) 󸀠󸀠 𝑃2󸀠󸀠 (𝑥𝑛+𝑘 ) ⋅ ⋅ ⋅ 𝑃𝑘+3 (𝑥𝑛+𝑘 ) ) where 𝑃𝑖 (𝑥) = 𝑥𝑖−1 , 𝑖 = 1, 2, . . . , 𝑘 + 1, 𝑃𝑘+2 (𝑥) = sin 𝑤𝑥, and 𝑃𝑘+3 (𝑥) = cos 𝑤𝑥. 2. Derivation of the Family of Methods Remark 1. In the derivation of the ESDM, the bases 𝑃(𝑥) ≡ 𝑃𝑖 (𝑥)𝑇 with 𝑃𝑖 (𝑥) = 𝑥𝑖−1 , 𝑖 = 1, 2, . . . , 𝑘+1, 𝑃𝑘+2 (𝑥) = sin(𝑤𝑥), and 𝑃𝑘+3 (𝑥) = cos(𝑤𝑥) are chosen because they are simple to analyze. Other possible bases (see Nguyen et al. [16] and Nguyen et al. [23]) include the following: We consider the first-order differential equation 𝑦󸀠 = 𝑓 (𝑥, 𝑦) , 𝑦 (𝑎) = 𝑦0 , We now define the following vectors and matrix used in the following theorem: (2) 𝑥 ∈ [𝑎, 𝑏] , (1) {sin(𝑤𝑥), cos(𝑤𝑥), 𝑥sin(𝑤𝑥), 𝑥cos(𝑤𝑥), . . ., 𝑥𝑛 sin(𝑤𝑥), 𝑥𝑛 cos(𝑤𝑥)}; (2) {sin𝑥, cos𝑥, . . . , sin(𝑛𝑤𝑥), cos(𝑛𝑤𝑥)}; where 𝑓 is assumed to satisfy the conditions to guarantee the existence of a unique solution of the initial-value problem. 2.1. CSDMM. In what follows, we state the CSDMM which has the ability to produce the ESDM: (4) {𝑥, . . . , 𝑤𝑥𝑛 } cos(𝑚𝑤𝑥)}; ∪ {sin(𝑤𝑥), cos(𝑤𝑥), . . . , sin(𝑚𝑤𝑥), (5) {𝑥, . . . , 𝑥𝑛 , exp(±𝑤𝑥), 𝑥exp(±𝑤𝑥), . . . , 𝑥𝑚 exp(±𝑤𝑥)}; (6) {𝑥, . . . , 𝑤𝑥𝑛−1 , 𝑤𝑥𝑛 }. Theorem 2. Let 𝑈(𝑥) satisfy 𝑈(𝑥𝑛+𝑗 ) = 𝑦𝑛+𝑗 , 𝑈󸀠 (𝑥𝑛+𝑗 ) = 𝑓𝑛+𝑗 , and 𝑈󸀠󸀠 (𝑥𝑛+𝑗 ) = 𝑔𝑛+𝑗 and let 𝑊 be invertible; then method (3) is equivalent to 𝑘 𝑈 (𝑥) = 𝛼𝑛+𝑘−1 (𝑥) 𝑦𝑛+𝑘−1 + ℎ∑ 𝛽𝑗 (𝑥) 𝑓𝑛+𝑗 𝑗=0 (3) {sin(𝑤1 𝑥), cos(𝑤1 𝑥), . . . , sin(𝑤𝑛 𝑥), cos(𝑤𝑛 𝑥)}; (3) 2 + ℎ 𝛾𝑛+𝑘 (𝑥) 𝑔𝑛+𝑘 , 𝑇 𝑈 (𝑥) = 𝑉𝑇 (𝑊−1 ) 𝑃 (𝑥) . (5) The proof of the above theorem can be found in Jator et al. [24]. where 𝛼𝑛+𝑘−1 (𝑥), 𝛽𝑗 (𝑥), and 𝛾𝑛+𝑘 (𝑥) are continuous coefficients. We assume that 𝑦𝑛+𝑗 = 𝑈(𝑥𝑛+𝑗 ) is the numerical 󸀠 approximation to the analytical solution 𝑦(𝑥𝑛+𝑗 ), 𝑦𝑛+𝑗 = 𝑈󸀠 (𝑥𝑛+𝑗 ) is the numerical approximation to the analytical solution 𝑦󸀠 (...truncated)