Journal of Applied Mathematics

Volume 2015, Article ID 343295, 17 pages

http://dx.doi.org/10.1155/2015/343295

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

^{1}Department of Mathematics, USC Salkehatchie, Walterboro, SC 29488, USA^{2}Department of Mathematics and Statistics, Austin Peay State University, Clarksville, TN 37044, USA

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.

#### Abstract

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:where and is 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 Runge-Kutta 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 for a step method.

In this paper, we propose a numerical integration formula which more effectively copes with stiff and/or oscillatory 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 , an approximation is obtained at only after an approximation at has been computed, where , , , , is the step size, is a positive integer, and 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.

#### 2. Derivation of the Family of Methods

We consider the first-order differential equationwhere 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:where , , and are continuous coefficients. We assume that is the numerical approximation to the analytical solution , is the numerical approximation to the analytical solution , is an approximation to , and is an approximation to , where , , .

We now define the following vectors and matrix used in the following theorem:where , , , and .

*Remark 1. *In the derivation of the ESDM, the bases with , , , and are chosen because they are simple to analyze. Other possible bases (see Nguyen et al. [16] and Nguyen et al. [23]) include the following: (1);(2);(3);(4);(5);(6).

Theorem 2. *Let satisfy , , and and let be invertible; then method (3) is equivalent to**The proof of the above theorem can be found in Jator et al. [24].*

Through interpolation of at the point , collocation of at the points , , and collocation of at the point , we get the systemTo solve this system we require that method (3) be defined by the assumed basis functions where the constants , , and are to be determined.

##### 2.2. ESDM

The general second derivative formula for solving (2) using the -step second derivative linear multistep method is of the formwhere , and , ; is a discrete point at node ; and , , and are parameters to be determined. It is worth noting that Enright’s method is a special case of (7). We solve (6) to get the coefficients , , and in (7) which are then used to obtain the continuous multistep method of Enright in the formEvaluating (9) at and setting yield the following Enright’s second derivative multistep method:whereas the complementary methodsare obtained by evaluating (9) at , , with .

We note that, in order to avoid the cancellations which might occur when is small, the use of the power series expansions of , , , and is preferable (see Simos [25]).

*Case *. This case has only the main method given by (10) with the coefficients defined by

*Case *. The coefficients of the main method (10) and the complimentary method (11) are, respectively, defined by

*Case *. The coefficients of the main method (10) and the complimentary methods (11) are, respectively, defined by

*Case *. The coefficients of the main method (10) and the complimentary methods (11) are, respectively, defined bywith , , and defined in part A of Appendix of the supplementary material (see Supplementary Material available online at http://dx.doi.org/10.1155/2015/343295). Consider with , , and defined in part B of Appendix of the supplementary material. Considerwith , , defined in part C of Appendix of the supplementary material. Considerwith , , and defined in part D of Appendix of the supplementary material.

##### 2.3. Block Specification and Implementation of the Methods

We consider a general procedure for the block implementation of the methods in matrix form (see Fatunla [26]). First we define the following vectors:where , and . The integration on the entire block will be compactly written aswhich forms a nonlinear equation because of the implicit nature, and hence we employ the Newton iteration for the evaluation of the approximate solutions. We use Newton’s approach for the implementation of implicit schemes to get the following solution of the block:The matrices , , , , and are defined as follows.

*Case *. Considerwith , , , , , , defined in methods (13) and (14).

*Case **.* Considerwith , , , , , , defined in methods (15), (16), and (17).

*Case **.* Considerwith , , , , and , defined in methods (18) through (21).

#### 3. Error Analysis and Stability

##### 3.1. Local Truncation Error (LTE)

Suppose that method (10) is associated with a linear difference operator:where is an arbitrary smooth function. Then is called the local truncation error at if represents a solution of the IVP (2). By a Taylor series expansion of , , and , we have