Journal of Applied Mathematics

Volume 2013 (2013), Article ID 691614, 6 pages

http://dx.doi.org/10.1155/2013/691614

## Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

^{1}Department of Mathematics, Faculty of Science, Muğla University, 48000 Muğla, Turkey^{2}Department of Mathematics, Faculty of Science, Celal Bayar University, 45000 Manisa, Turkey

Received 30 December 2012; Revised 14 May 2013; Accepted 20 May 2013

Academic Editor: Han H. Choi

Copyright © 2013 Berna Bülbül and Mehmet Sezer. 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

We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems.

#### 1. Introduction

Nonlinear ordinary differential equations are frequently used to model a wide class of problems in many areas of scientific fields: chemical reactions, spring-mass systems bending of beams, resistor-capacitor-inductance circuits, pendulums, the motion of a rotating mass around another body, and so forth [1, 2]. Also, nonlinear equations which can be modeled by the oscillator equations are of crucial importance in all areas of engineering sciences [3–7]. Thus, methods of solution for these equations are of great importance to engineers and scientists. Many new techniques have appeared in the open literature such as homotopy perturbation transform method [8], variational iteration method [9], energy balance method [10], Hamiltonian approach [11], coupled homotopy-variational formulation [12], variational approach [13], and amplitude-frequency formulation [14].

In this paper, a new method is introduced for the following model of nonlinear problems: with the initial conditions where , , , , and are real constants.

Equation (1) has been discussed in many works [15–21], for different systems arising in various scientific fields such as physics, engineering, biology, and communication theory. Recently Wang [19] presented the quasi-two-step method for the nonlinear undamped Duffing equation. Donnagáin and Rasskazov [15] studied a modification of the Duffing equation describing a periodically driven iron pendulum in nonuniform magnetic field. Feng [16] illustrated a connection between the Duffing equation and Hirota equation and obtained two periodic wave solutions in terms of elliptic functions of the Hirota equation, by using the exact solution of Duffing equation. A direct method to find the exact solution to the damped Duffing equation and traveling wave solutions to the reaction-diffusion equation was used by Feng [17]. In addition, the solution of the Duffing equation in nonlinear vibration problem by using target function method was investigated by Chen [18]. The Laplace decomposition method for numerical solution of Duffing equation has been introduced by Yusufoğlu [20] and Khuri [21]. On the other hand, Duffing differential equations have also been effectively dealt in many works [22–29].

The aim of this study is to get solution as truncated Taylor series centered about zero defined by which is Taylor polynomial of degree at , where , are unknown Taylor coefficients to be determined.

#### 2. Fundamental Relations

In this section we convert the expressions defined in (1) and (2) to the matrix forms by the following procedure. Firstly, the function defined by (3) can be written in the matrix form where

On the other hand, it is clearly seen that the relation between the matrix and its derivative is where From the matrix equations (4) and (6), it follows that By using the production of two series, the matrix form of expression is obtained as where Similarly, the matrix representation of becomes where Finally, by substituting the matrix forms (4)–(11) into (1) we have the fundamental matrix equation

#### 3. Method of the Solution

To obtain the Taylor polynomial solution of (1) in the form (3) we firstly compute the Taylor coefficients by means of the collocation points defined by By substituting the collocation points into matrix equation (13) we obtain the system of matrix equations or compact notation Briefly, we can write the matrix equation which corresponds to a system of nonlinear algebraic equations with the unknown Taylor coefficients , . The matrices in (17) are as follows: where The augmented matrix for (17) becomes, clearly,

Next, by means of relations (4) and (8), we can obtain the corresponding matrix forms for the initial conditions (2) as

Consequently, to obtain the solution of (1) under the conditions (2), by replacing the row matrices (21) by the last two rows of the augmented matrix (20), we have the required augmented matrix

or By solving the matrix equation (22), the unknown Taylor coefficients are determined. Thus we get the Taylor polynomial solution

#### 4. Error Bound and Accuracy of the Solution

We can easily check the accuracy of the solution as follows. Since the Taylor polynomial (3) is an approximate solution of (1), then the solution is substituted in (1), and the resulting equation must be satisfied approximately; that is, for , , or ( positive integer).

If max ( positive integer) is prescribed, then the truncation limit is increased until the difference at each of the points becomes smaller than the prescribed . On the other hand, the error can be estimated by means of the function If , when is sufficiently large enough, then the error decreases.

Also if we know the exact solution of problem, we can find error bound of method.

Theorem 1 (Lagrange error bound). * Let be function such that it and all of its derivatives are continuous, and is the th Taylor polynomial for centered ; then the error is bounded by
**
where , . *

If is approximate solution of the matrix method and is the th Taylor polynomial for , we can write , and absolute error function of the method is . To obtain error bound by Theorem 1 we have Therefore we can write which gives us error bound.

#### 5. Numerical Examples

The method of this study is useful in finding the solutions of Duffing equations in terms of Taylor polynomials. We illustrate it by the following examples. Numerical computations have been done using Maple 9.

*Example 1. *Consider the Duffing equation in the following type [20]:
with the initial conditions

We assume that the problem has a Taylor polynomial solution in the form

The solutions obtained for are compared with the exact solution [20] (see Table 1 and Figure 1). We calculate the following Lagrange error bound for different values of , respectively, which is 2 decimal places accuracy, which is 5 decimal places accuracy, and which is 7 decimal places accuracy.

*Example 2. * Consider the Duffing equation [20]
with initial conditions

Applying the present method, we have the following fundamental matrix equation:

The solution of this nonlinear system is obtained for , and 20. For numerical results, see Tables 2, 3 and Figure 2.

*Example 3. * In this last example, the method is illustrated by considering the damped Duffing equation [21]
Consider the case .

We have similar results for Example 3 as in Example 2. Table 4 shows the error values for , and .

#### 6. Conclusion

In this paper, a very simple but effective Taylor matrix method was proposed for the numerical solution of the nonlinear Duffing equation. One of the advantages of this method that is the solution is expressed as a truncated Taylor series, then can be easily evaluated for arbitrary values of by using the computer program without any computational effort. From the given illustrative examples, it can be seen that the Taylor series approach can obtain very accurate and satisfactory results. However, the main point is that the polynomial expansion is highly ill-posed when is larger than 1. This method can be improved with new strategies to solve the other nonlinear equations.

#### References

- G. F. Corliss,
*Guarented Error Bounds for Ordinary Differential Equations, in Theory and Numeric of Ordinary and Partial Equations*, Oxford University press, Oxford, UK, 1995. - H. Bulut and D. J. Evans, “On the solution of the Riccati equation by the decomposition method,”
*International Journal of Computer Mathematics*, vol. 79, no. 1, pp. 103–109, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Khan, M. Mada, A. Yildirim, M. A. Abdou, and N. Faraz, “A new approach to van der Pol’s oscillator problem,”
*Zeitschrift für Naturforschung A*, vol. 66, pp. 620–624, 2011. View at Google Scholar - Y. Khan, H. Vázquez-Leal, and N. Faraz, “An efficient new iterative method for oscillator differential equation,”
*Scientia Iranica*, vol. 19, pp. 1473–1477, 2012. View at Google Scholar - Y. Khan, “Mehdi Akbarzade Dynamic analysis of non-linear oscillator equation arising in double-sided driven clamped microbeam-based electromechanical resonator,”
*Zeitschrift für Naturforschung A*, vol. 67, pp. 435–440, 2012. View at Google Scholar - Y. Khan, M. Akbarzade, and A. Kargar, “Coupling of homotopy and variational approach for conservative oscillator with strong odd-nonlinearity,”
*Scientia Iranica*, vol. 19, pp. 417–422, 2012. View at Google Scholar - M. Akbarzade and Y. Khan, “Dynamic model of large amplitude non-linear oscillations arising in the structural engineering: analytical solutions,”
*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 480–489, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Khan and Q. Wu, “Homotopy perturbation transform method for nonlinear equations using He's polynomials,”
*Computers & Mathematics with Applications*, vol. 61, no. 8, pp. 1963–1967, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. H. He, G. C. Wu, and F. Austin, “The variational iteration method which should be followed,”
*Nonlinear Science Letters A*, vol. 1, pp. 1–30, 2010. View at Google Scholar - N. Jamshidi and D. D. Ganji, “Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire,”
*Current Applied Physics*, vol. 10, pp. 484–486, 2010. View at Google Scholar - J.-H. He, “Hamiltonian approach to nonlinear oscillators,”
*Physics Letters A*, vol. 374, no. 23, pp. 2312–2314, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Akbarzade and J. Langari, “Determination of natural frequencies by coupled method of homotopy perturbation and variational method for strongly nonlinear oscillators,”
*Journal of Mathematical Physics*, vol. 52, no. 2, Article ID 023518, 10 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - J.-H. He, “Variational approach for nonlinear oscillators,”
*Chaos, Solitons & Fractals*, vol. 34, no. 5, pp. 1430–1439, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. D. Ganji and M. Akbarzade, “Approximate analytical solutions to non-linear oscillators using He’s amplitude-frequency formulation,”
*International Journal of Mathematical Analysis*, vol. 32, no. 4, pp. 1591–1597, 2010. View at Google Scholar - M. O. Donnagáin and O. Rasskazov, “Numerical modelling of an iron pendulum in a magnetic field,”
*Physica B*, vol. 372, no. 1-2, pp. 37–39, 2006. View at Google Scholar - Z. Feng, “Duffing's equation and its applications to the Hirota equation,”
*Physics Letters A*, vol. 317, no. 1-2, pp. 115–119, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Feng, “Monotonous property of non-oscillations of the damped Duffing's equation,”
*Chaos, Solitons and Fractals*, vol. 28, no. 2, pp. 463–471, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Z. Chen, “Solution of the Duffing equation by using target function method,”
*Journal of Sound and Vibration*, vol. 256, no. 3, pp. 573–578, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Wang, “P-stable linear symmetric multistep methods for periodic initial-value problems,”
*Computer Physics Communications*, vol. 171, no. 3, pp. 162–174, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Yusufoğlu, “Numerical solution of Duffing equation by the Laplace decomposition algorithm,”
*Applied Mathematics and Computation*, vol. 177, no. 2, pp. 572–580, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. A. Khuri, “A Laplace decomposition algorithm applied to a class of nonlinear differential equations,”
*Journal of Applied Mathematics*, vol. 1, no. 4, pp. 141–155, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Pezeshki and E. H. Dowell, “An examination of initial condition maps for the sinusoidally excited buckled beam modeled by the Duffing's equation,”
*Journal of Sound and Vibration*, vol. 117, no. 2, pp. 219–232, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. A. Khuri and S. Xie, “On the numerical verification of the asymptotic expansion of Duffing's equation,”
*International Journal of Computer Mathematics*, vol. 372, no. 3, pp. 25–30, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. H. Ku and X. Sun, “CHAOS and limit cycle in Duffing's equation,”
*Journal of the Franklin Institute*, vol. 327, no. 2, pp. 165–195, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Battelli and K. J. Palmer, “Chaos in the Duffing equation,”
*Journal of Differential Equations*, vol. 101, no. 2, pp. 276–301, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Loria, E. Panteley, and H. Nijmeijer, “Control of the chaotic Duffing equation with uncertainty in all parameters,”
*IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications*, vol. 45, no. 12, pp. 1252–1255, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Nijmeijer and H. Berghuis, “On Lyapunov control of the Duffing equation,”
*IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications*, vol. 42, no. 8, pp. 473–477, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. G. Byatt-Smith, “Regular and chaotic solutions of Duffing's equation for large forcing,”
*IMA Journal of Applied Mathematics*, vol. 37, no. 2, pp. 113–145, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Habets and G. Metzen, “Existence of periodic solutions of Duffing equations,”
*Journal of Differential Equations*, vol. 78, no. 1, pp. 1–32, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet