Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 217578, 11 pages

http://dx.doi.org/10.1155/2015/217578

## Optimized Hybrid Methods for Solving Oscillatory Second Order Initial Value Problems

^{1}Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia^{2}Department of Mathematics, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Received 25 October 2014; Accepted 6 January 2015

Academic Editor: Xiaohua Ding

Copyright © 2015 N. Senu 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.

#### Abstract

Two-step optimized hybrid methods of order five and order six are developed for the integration of second order oscillatory initial value problems. The optimized hybrid method (OHMs) are based on the existing nonzero dissipative hybrid methods. Phase-lag, dissipation or amplification error, and the differentiation of the phase-lag relations are required to obtain the methods. Phase-fitted methods based on the same nonzero dissipative hybrid methods are also constructed. Numerical results show that OHMs are more accurate compared to the phase-fitted methods and some well-known methods appeared in the scientific literature in solving oscillating second order initial value problems. It is also found that the nonzero dissipative hybrid methods are more suitable to be optimized than phase-fitted methods.

#### 1. Introduction

Many differential equations which appear in practice are system of second-order initial value problem (IVP), in which the first derivative does not appear explicitly: This type of ordinary differential equations (ODEs) often appears in many scientific areas of engineering and applied sciences such as celestial mechanics, molecular dynamics, quantum mechanics, and theoretical physics. Quite often the solution of (1) exhibits a pronounced oscillatory character. Oscillatory problems are usually harder to solve than the nonoscillatory problems. Coleman [1] in his paper developed the order conditions of hybrid method up to order nine. With the order condition developed by Coleman [1], Franco [2] constructed explicit two-step hybrid methods of fourth, fifth, and sixth order for solving second order IVPs.

To obtain a more efficient process for solving oscillatory problems, numerical methods are constructed by taking into account the nature of the problem. This results in methods in which the coefficients depend on the problem. Some important classes of numerical methods, such as exponentially fitted or phase-fitted methods, can be obtained if a good estimate of the period is known.

Phase-lag or dispersion error is the angle between the true and the approximated solutions. The analysis of phase-lag or dispersion error was first introduced by Brusa and Nigro [3]. Several authors such as van der Houwen and Sommeijer [4, 5] and Thomas [6] studied in detail the phase-lag of numerical methods for solving (1). Several authors in their papers [6–9] have developed hybrid methods with the purpose of making the phase-lag of the method smaller. The technique of vanishing the phase-lag, the amplification error, and their first integral of phase-lag was introduced by Papadopoulos and Simos [10]. Numerical results indicated that such technique produced methods which are very efficient for solving Schrodinger equation. Kosti et al. [11] in their work developed an optimized explicit Runge-Kutta Nyström method with four stages and fifth algebraic order for the numerical solution of orbital and related periodic initial value problems. Other than phase-lag much research is also focused on methods having high dissipative order; dissipation is the distance of the computed solution from the standard cyclic solution. Hence, for solving oscillatory problems, it is the aim of every researcher to derive numerical method which has high algebraic order, phase-lag order, and dissipative order. Methods having dissipative order infinity are called a zero-dissipative methods.

In this work, we will derive methods in which the coefficients depend on the problems. The first two methods are called optimized methods which are developed using the same approach introduced by Kosti et al. [11]. They are optimized by imposing the phase-lag, dissipative or amplification error, and the first derivative of the phase-lag relation. The second two methods derived are called phase-fitted methods which are obtained by minimizing the phase-lag. The new optimized and phase-fitted hybrid methods are based on the nonzero-dissipative, four-stage fifth-order and five-stage sixth-order hybrid methods developed by Franco [2].

The comparison of the new methods with other methods in the scientific literature has shown that the optimized methods are more efficient for solving problems of oscillating in nature.

#### 2. Analysis Phase-Lag of the Methods

An -stage two-step hybrid method for numerical integration of the IVP (1) is in the form where , , and can be represented in Butcher notation by the table of coefficients.

-stage hybrid methods are as follows:

The methods of the form (2) are defined by where , , and and the first two nodes are and . The method only requires to evaluate , for each step after starting procedure. This method is considered as two-step hybrid method with stages per step and can be written in Butcher tableau; -stage explicit hybrid methods are as follows:

In order to construct the new method, we use the test equation

By replacing into (2) we have

Define , so (7) can also be written as

Alternatively (8) can be written in vector form as follows: where , , ,

From (9) we obtain

Substituting (12) into (10) gives

Rewrite (13) and then the following recursion is obtained where

The following definition is given by van der Houwen and Sommeijer [4] for Runge-Kutta Nyström method and has been used by Franco [2] for hybrid method.

*Definition 1. *Apply the hybrid method (2) to (6). Next we define the phase-lag, and dissipation, and
If , then the hybrid method is said to have phase-lag order . If , then the hybrid method is said to have dissipation order .

#### 3. Derivation of the New Hybrid Methods

In this section, we construct optimized hybrid methods of four-stage fifth order and five-stage sixth order. First, let us denote (15) in polynomials form as follows:

The relations between phase-lag (dispersion), dissipation (amplification error), and the derivative of phase-lag are taken into consideration throughout obtaining the new methods. On the other hand, phase-fitted methods which are based on the same hybrid methods are also developed. The effectiveness of the new optimized methods is compared with that of the phase-fitted method as well as other existing methods in the literature.

##### 3.1. Optimized Hybrid Methods

Polynomials (17) for hybrid method which satisfied algebraic order conditions up to order six can be written in these expressions:(i)for , (ii)for , In order to develop optimized hybrid method the following relations must hold: the phase-lag condition, dissipation condition, and the first derivative of (20),

The hybrid method in Franco (2006) [2], which is four-stage fifth order with dispersion order eight and dissipation order five or which we called nonzero dissipative method since the order of the dissipation is finite, is then substituted into the nullifying equations (20)–(22), and the equations are solved numerically. The coefficients of the method with , , and taken as free parameters can be written in Butcher tableau as follows (see (26)).

The optimized hybrid method is obtained with , , and given by

An optimized four stage fifth-order hybrid method is as follows: and the free parameters in Taylor expansion are given by

For the construction of five-stage sixth-order optimized hybrid methods, we substitute (18) into (20)–(22) and using the coefficients of the five-stage sixth-order method which is nonzero dissipative given in Franco (2006) [2] (see (30)).

An optimized five-stage sixth-order hybrid method is as follows:

Choosing , , and as the free parameters, the optimized sixth-order hybrid method is obtained: and the free parameters in Taylor expansion are given as follows:

##### 3.2. Phase-Fitted Hybrid Methods

In this paper, we also develop the phase-fitted the original hybrid methods. To develop phase-fitted method, (20) must hold. Equation (18) is substituted into (20), choosing as the free parameter and using the same coefficients as in (26), together with

We obtain a phase-fitted hybrid method of four-stage fifth-order with the solution of which is given by and the Taylor expansion of the free parameter is

For the construction of five-stage sixth-order phase-fitted hybrid method, we substitute (19) into (20) and set as the free parameter. We use the coefficients given in (30) together with

Choosing as the free parameter, we get a phase-fitted hybrid method of five-stage sixth-order and is given by and the Taylor expansion of is

#### 4. Problems Tested and Numerical Results

In this section, we compare the optimized methods of order five and order six with the phase-fitted methods together with a few existing methods in the literature in order to determine the accuracy of the new optimized methods. The existing methods are the original hybrid method by Franco [2], optimized Runge-Kutta Nyström method order five developed by Kosti et al. [11], phase-fitted Runge-Kutta Nyström method of order four developed by Papadopoulos et al. [12], Runge-Kutta Nyström method of order five by Hairer et al. [13], Runge-Kutta method of order six given in Butcher [14], and phase-fitted and amplification-fitted Runge-Kutta Nyström method of order four developed in Papadopoulos et al. [12]. Efficiency curves of the fifth and sixth order methods are given in Figures 1–7 and 8–14, respectively. The test problems are listed below.