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

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.

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Berna Bülbül 1 and Mehmet Sezer 2

Recommended by Han H. Choi

1, Department of Mathematics, Faculty of Science, Mugla University, 48000 Mugla, 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

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.

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: [figure omitted; refer to PDF] with the initial conditions [figure omitted; refer to PDF] where p , _p1 , _p2 , α , and β 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 Yusufoglu [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 [figure omitted; refer to PDF] which is Taylor polynomial of degree N at x=0 , where _yn , n=0,1,...,N 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 y(x) defined by (3) can be written in the matrix form [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

On the other hand, it is clearly seen that the relation between the matrix X(x) and its derivative ^{X[variant prime]} (x) is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] From the matrix equations (4) and (6), it follows that [figure omitted; refer to PDF] By using the production of two series, the matrix form of expression ^y2 (x) is obtained as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Similarly, the matrix representation of ^y3 (x) becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Finally, by substituting the matrix forms (4)-(11) into (1) we have the fundamental matrix equation [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] By substituting the collocation points into matrix equation (13) we obtain the system of matrix equations [figure omitted; refer to PDF] or compact notation [figure omitted; refer to PDF] Briefly, we can write the matrix equation [figure omitted; refer to PDF] which corresponds to a system of (N+1) nonlinear algebraic equations with the unknown Taylor coefficients _yn , n=0,1,...,N . The matrices in (17) are as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The augmented matrix for (17) becomes, clearly, [figure omitted; refer to PDF]

Next, by means of relations (4) and (8), we can obtain the corresponding matrix forms for the initial conditions (2) as [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF]

or [figure omitted; refer to PDF] By solving the matrix equation (22), the unknown Taylor coefficients _yn are determined. Thus we get the Taylor polynomial solution [figure omitted; refer to PDF]

4. Error Bound and Accuracy of the Solution

We can easily check the accuracy of the solution _yN (x) as follows. Since the Taylor polynomial (3) is an approximate solution of (1), then the solution _yN (x) is substituted in (1), and the resulting equation must be satisfied approximately; that is, for x=_xi ∈[a,b] , i=0,1,2,... , [figure omitted; refer to PDF] or _EN (_xi )...4;1^0-_ki (_ki positive integer).

If max (1^0_ki )=1^0-k (k positive integer) is prescribed, then the truncation limit N is increased until the difference _EN (_xi ) at each of the points becomes smaller than the prescribed 1^0-k . On the other hand, the error can be estimated by means of the function [figure omitted; refer to PDF] If _EN (x)[arrow right]0 , when N 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 f be function such that it and all of its derivatives are continuous, and _fn (x) is the n th Taylor polynomial for f(x) centered x=a ; then the error is bounded by [figure omitted; refer to PDF] where M=max|^f(n+1) (ξ)| , ξ∈(a,x) .

If _yN is approximate solution of the matrix method and _fN (x) is the N th Taylor polynomial for f(x) , we can write |_fN -_yN |=[straight epsilon] , and absolute error function of the method is |f-_yN |=_eN . To obtain error bound by Theorem 1 we have [figure omitted; refer to PDF] Therefore we can write [figure omitted; refer to PDF] 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]: [figure omitted; refer to PDF] with the initial conditions [figure omitted; refer to PDF]

We assume that the problem has a Taylor polynomial solution in the form [figure omitted; refer to PDF]

The solutions obtained for N=5,8,and 10 are compared with the exact solution y(x)=sin(x) [20] (see Table 1 and Figure 1). We calculate the following Lagrange error bound for different values of N , respectively, _M5 =0.161×1^0-2 which is 2 decimal places accuracy, _M8 =0.352×1^0-5 which is 5 decimal places accuracy, and _M10 =0.248×1^0-7 which is 7 decimal places accuracy.

Table 1: Comparison of the absolute errors of Example 1.

x r	Exact	Error analysis (|y-yN | )
N = 5	N = 8	N = 10
0	0	0	0	0
0.1	0.9983341665	4.62 E - 8	3.08 E - 11	4.33 E - 14
0.2	0.1986693308	6.12 E - 7	8.72 E - 11	1.03 E - 13
0.3	0.2955202067	4.28 E - 7	1.45 E - 11	1.66 E - 13
0.4	0.3894183423	2.29 E - 7	1.82 E - 10	2.21 E - 13
0.5	0.4794255386	4.23 E - 7	1.65 E - 10	2.71 E - 13
0.6	0.5646424734	4.03 E - 7	2.62 E - 10	3.15 E - 13
0.7	0.6442176872	3.32 E - 7	3.01 E - 12	2.29 E - 13
0.8	0.7173560909	5.66 E - 7	3.07 E - 10	3.85 E - 13
0.9	0.7833260096	8.87 E - 6	5.68 E - 9	9.73 E - 13
1	0.841470984	1.43 E - 5	1.22 E - 8	1.57 E - 11

(a) Comparison of the approximate solution with exact solution for N=10 . (b) Comparison of absolute errors for N=5,8 , and 10.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Example 2.

Consider the Duffing equation [20] [figure omitted; refer to PDF] with initial conditions [figure omitted; refer to PDF]

Applying the present method, we have the following fundamental matrix equation: [figure omitted; refer to PDF]

The solution of this nonlinear system is obtained for N=5,10,15 , and 20. For numerical results, see Tables 2, 3 and Figure 2.

Table 2: Numerical results of Example 2.

x r	Error analysis (EN (xr ) )
N = 5	N = 10	N = 15	N = 20
0	0	0	1.70 E - 8	0
0.1	4.62 E - 2	0	0	0
0.2	5.57 E - 24	0	1.05 E - 9	2.23 E - 25
0.3	2.24 E - 3	0	0	2.22 E - 25
0.4	0	3.22 E - 24	2.91 E - 11	9.80 E - 25
0.5	6.00 E - 25	0	0	1.00 E - 24
0.6	0	2.51 E - 24	1.62 E - 11	1.70 E - 24
0.7	3.63 E - 2	4.31 E - 24	0	1.31 E - 24
0.8	1.13 E - 1	1.87 E - 4	1.80 E - 24	2.10 E - 24
0.9	1.46 E - 1	8.28 E - 3	1.32 E - 7	8.50 E - 24
1	5.93 E - 1	1.32 E - 2	4.47 E - 5	3.68 E - 24

Table 3: Comparison of numerical solutions.

x r	y ~ ( x ) (Laplace method)	y N ( x ) (present method)
0	0.3	0.3
0.1	0.0835845348	0.08358453484
0.2	-0.105091915	-0.1050919158
0.3	-0.266020303	-0.2660203036
0.4	-0.399977948	-0.3999779482
0.5	-0.508314810	-0.5083148108
0.6	-0.592890689	-0.5928906891
0.7	-0.656065502	-0.6560655028
0.8	-0.700676939	-0.7006769399
0.9	-0.729970850	-0.7299708508
1	-0.747476077	-0.7474760775

Figure 2: Error functions for N=10,15,and 20 of Example 2.

[figure omitted; refer to PDF]

Example 3.

In this last example, the method is illustrated by considering the damped Duffing equation [21] [figure omitted; refer to PDF] Consider the case α=β=k=1 .

We have similar results for Example 3 as in Example 2. Table 4 shows the error values _EN (_xr ) for N=5,8,10 , and 20 .

Table 4: Comparison of the absolute errors of Example 3.

x r	Error analysis (EN (xr ) )
N = 5	N = 8	N = 10	N = 20
0	0	0	0	0
0.1	4.62 E - 3	5.08 E - 6	0	0
0.2	0	6.72 E - 6	0	0
0.3	4.54 E - 3	1.45 E - 6	0	0
0.4	0	3.82 E - 6	0	0
0.5	7.85 E - 3	0	0	0
0.6	0	2.62 E - 5	0	0
0.7	5.42 E - 2	3.01 E - 5	0	0
0.8	1.94 E - 1	8.07 E - 4	0	0
0.9	0.46 E - 1	5.68 E - 3	0	0
1	0.93 E - 1	1.22 E - 2	4.57 E - 8	3.32 E - 29

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 y(x) can be easily evaluated for arbitrary values of x 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 x is larger than 1. This method can be improved with new strategies to solve the other nonlinear equations.

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