## Abstract

In this paper, a new perturbation technique is employed to solve strongly nonlinear Duffing oscillators, in which a new parameter ${\textstyle \alpha =\alpha (\epsilon )}$ is defined such that the value of α   is always small regardless of the magnitude of the original parameter ${\textstyle \epsilon }$. Therefore, the strongly nonlinear Duffing oscillators with large parameter ε   are transformed into a small parameter system with respect to ${\textstyle \alpha }$. Approximate solution obtained by the present method is compared with the solution of energy balance method, homotopy perturbation method, global error minimization method and lastly numerical solution. We observe from the results that this method is very simple, easy to apply, and gives a very good accuracy not only for small parameter εbut also for large values of ε.

## Keywords

Analytical solution; Perturbation technique; Strongly nonlinear Duffing oscillators

## 1. Introduction

Duffing oscillators are described by nonlinear differential equations that modeled the behavior of many practical problems that arise in engineering, physics, and in many real world applications [1], [2], [3], [4] and [5]. It is well known that Duffing oscillators can be found in the modeling of free vibrations of a restrained uniform beam with intermediate lumped mass, the nonlinear dynamics of slender elastica, the generalized Pochhammer–Chree (PC) equation, the generalized compound KdV equation in nonlinear wave systems, among others [6].

The study of Duffing oscillators has received considerable attention in recent years due to a variety of engineering applications. Several approaches have been proposed so far dealing with the nonlinear Duffing oscillators. Variational iteration method [7], homotopy perturbation method [8], He’s energy balance method [9], He’s parameter-expanding method [10], He’s max–min approach [11] and global error minimization method [12] are some examples.

In this paper, we will apply He’s modified perturbation technique [13] and [14] to solve nonlinear Duffing oscillators of fifth order in two cases first, without forced term and second with forced term, which hold for all the values of amplitude of the oscillator [15], [16] and [17].

## 2. Basic idea of the method

To illustrate the basic idea of the present note, we will consider the following nonlinear Duffing equation with nonlinearity of fifth order:

 ${\displaystyle {\ddot {u}}+u+\epsilon u^{5}=0{\mbox{,}}\quad u(0)=A{\mbox{,}}\quad {\overset {\cdot }{u}}(0)=}$${\displaystyle 0{\mbox{.}}}$
(1)

Re-write Eq. (1) in the form

 ${\displaystyle {\ddot {u}}+{\beta }^{2}u+\epsilon (\eta u+u^{5})=0{\mbox{,}}\quad u(0)=}$${\displaystyle A{\mbox{,}}\quad {\overset {\cdot }{u}}(0)=0{\mbox{,}}}$
(2)

where

 ${\displaystyle {\beta }^{2}+\epsilon \eta =1{\mbox{,}}\quad \epsilon \eta <1{\mbox{.}}}$
(3)

Here β is the angular frequency which is unknown to be further determined and ε is a large parameter.

Now define a new parameter

 ${\displaystyle \alpha ={\frac {\epsilon }{1+\epsilon }}{\mbox{,}}}$
(4)

such that

 ${\displaystyle \epsilon ={\frac {\alpha }{1-\alpha }}{\mbox{.}}}$
(5)

Then ${\textstyle \alpha \in \left[0{\mbox{,}}1\right)}$, when ${\textstyle \epsilon \in \left[0{\mbox{,}}\infty \right)}$, so ε can be written as follows:

 ${\displaystyle \epsilon =\alpha (1+\alpha +{\alpha }^{2}+{\alpha }^{3}+}$${\displaystyle \cdots ){\mbox{.}}}$
(6)

Assume that η and u can be written as follows:

 ${\displaystyle \eta ={\eta }_{0}+\alpha {\eta }_{1}+{\alpha }^{2}{\eta }_{2}+}$${\displaystyle \cdots {\mbox{.}}}$
(7)

 ${\displaystyle u=u_{0}+\alpha u_{1}+{\alpha }^{2}u_{2}+\cdots {\mbox{.}}}$
(8)

Substituting Eqs. (6), (7) and (8) into Eq. (2) and equating coefficients of like powers of α yield the following equations:

 ${\displaystyle {\ddot {u}}_{0}+{\beta }^{2}u_{0}=0{\mbox{,}}\quad u_{0}(0)=A{\mbox{,}}\quad {\overset {\cdot }{u}}_{0}(0)=}$${\displaystyle 0{\mbox{,}}}$
(9)

 ${\displaystyle {\ddot {u}}_{1}+{\beta }^{2}u_{1}+{\eta }_{0}u_{0}+u_{0}^{5}=0{\mbox{,}}\quad u_{1}(0)=}$${\displaystyle 0{\mbox{,}}\quad {\overset {\cdot }{u}}_{1}(0)=0{\mbox{,}}}$
(10)

 ${\displaystyle {\ddot {u}}_{2}+{\beta }^{2}u_{2}+{\eta }_{1}u_{0}+{\eta }_{0}u_{1}+}$${\displaystyle {\eta }_{0}u_{0}+5u_{0}^{4}u_{1}+u_{0}^{5}=0{\mbox{,}}\quad u_{2}(0)=}$${\displaystyle 0{\mbox{,}}\quad {\overset {\cdot }{u}}_{2}(0)=0{\mbox{.}}}$
(11)

Solving Eqs. (9), (10) and (11), we obtain,

 ${\displaystyle u_{0}=Acos\beta t{\mbox{,}}\quad {\eta }_{0}=-{\frac {5A^{4}}{8}}{\mbox{,}}}$
(12)
 ${\displaystyle u_{1}={\frac {5A^{5}}{128{\beta }^{2}}}cos3\beta t+{\frac {A^{5}}{384{\beta }^{2}}}cos5\beta t{\mbox{,}}\quad {\eta }_{1}=}$${\displaystyle -{\frac {95A^{8}}{1536{\beta }^{2}}}}$
(13)
 ${\displaystyle u_{2}=\left({\frac {320A^{9}}{49152{\beta }^{4}}}+{\frac {5A^{5}}{128{\beta }^{2}}}\right)cos3\beta t+}$${\displaystyle \left({\frac {320A^{9}}{147456{\beta }^{4}}}+{\frac {A^{5}}{384{\beta }^{2}}}\right)cos5\beta t+}$${\displaystyle {\frac {95A^{9}}{294912{\beta }^{4}}}cos7\beta t+{\frac {5A^{9}}{491520{\beta }^{4}}}cos9\beta t{\mbox{.}}}$
(14)

Substituting ${\textstyle {\eta }_{0}}$ and ${\textstyle {\eta }_{1}}$ into Eq. (3), we have

 ${\displaystyle {\beta }^{4}-\left(1+{\frac {5\epsilon A^{4}}{8}}\right){\beta }^{2}-}$${\displaystyle {\frac {95\epsilon \alpha A^{8}}{1536}}=0{\mbox{.}}}$
(15)

Then, the second approximate solution to Eq. (1) becomes,

 ${\displaystyle u(t)=Acos\beta t+\alpha \left({\frac {5A^{5}}{128{\beta }^{2}}}cos3\beta t+\right.}$${\displaystyle \left.{\frac {A^{5}}{384{\beta }^{2}}}cos5\beta t\right)+}$${\displaystyle {\alpha }^{2}\left\{\left({\frac {320A^{9}}{49152{\beta }^{4}}}+\right.\right.}$${\displaystyle \left.\left.{\frac {5A^{5}}{128{\beta }^{2}}}\right)cos3\beta t+\right.}$${\displaystyle \left.\left({\frac {320A^{9}}{147456{\beta }^{4}}}+{\frac {A^{5}}{384{\beta }^{2}}}\right)cos5\beta t+\right.}$${\displaystyle \left.{\frac {95A^{9}}{294912{\beta }^{4}}}cos7\beta t+{\frac {5A^{9}}{491520{\beta }^{4}}}cos9\beta t\right\}{\mbox{.}}}$
(16)

To show the remarkable accuracy of the obtained result, the approximate solution with the energy balance method [18], homotopy perturbation method [19], global error minimization method [20] and numerical solution are compared at ${\textstyle \epsilon =1}$ in Table 1 (see Fig. 1).

Table 1. Comparison of the present method with the energy balance method, homotopy perturbation method, global error minimization method and numerical solution.
A EBM [18] HPM [19] GEMM [20] Present Numerical
0.1 1.00003 1.00003 1.00003 1.00003 1.00003
0.2 1.00047 1.00050 1.00050 1.00050 1.00050
0.3 1.00236 1.00253 1.00254 1.00253 1.00253
0.5 1.01807 1.01934 1.01941 1.01940 1.01940
1 1.25831 1.27475 1.28082 1.28211 1.28079
5 19.1202 19.7895 20.3919 20.5027 19.9719
10 76.3828 79.0633 81.4807 81.9253 80.0801

 Figure 1. Comparison of the approximate solution (solid line) with the numerical solution (dashed line).

## 3. Forced Duffing equation of fifth power nonlinear function

We consider next an oscillator with fifth power nonlinearity in the form

 ${\displaystyle {\ddot {u}}+u+\epsilon u^{5}=pcos\Omega t{\mbox{,}}\quad u(0)=}$${\displaystyle A{\mbox{,}}\quad {\overset {\cdot }{u}}(0)=0{\mbox{,}}}$
(17)

which represents a forced Duffing equation. Changing Eq. (17) into another case by using Eq. (3) to obtain

 ${\displaystyle {\ddot {u}}+{\beta }^{2}u+\epsilon (\eta u+u^{5})=pcos\Omega t{\mbox{.}}}$
(18)

Substituting Eqs. (6) and (8) into Eq. (18), and setting the coefficients of the powers of α equal to zero, resulting in

 ${\displaystyle {\ddot {u}}_{0}+{\beta }^{2}u_{0}=pcos\Omega t{\mbox{,}}\quad u_{0}(0)=}$${\displaystyle A{\mbox{,}}\quad {\overset {\cdot }{u}}_{0}(0)=0{\mbox{,}}}$
(19)

 ${\displaystyle {\ddot {u}}_{1}+{\beta }^{2}u_{1}+\eta u_{0}+u_{0}^{5}=0{\mbox{,}}\quad u_{1}(0)=}$${\displaystyle 0{\mbox{,}}\quad {\overset {\cdot }{u}}_{1}(0)=0{\mbox{.}}}$
(20)

Solving Eq. (19), results in

 ${\displaystyle u_{0}={\frac {p}{\left({\beta }^{2}-{\Omega }^{2}\right)}}cos\Omega t+}$${\displaystyle \left(A-{\frac {p}{{\beta }^{2}-{\Omega }^{2}}}\right)cos\beta t=}$${\displaystyle mcos\beta t+ncos\Omega t{\mbox{.}}}$
(21)

Substituting Eq. (21) into Eq. (20), to obtain

 ${\displaystyle \eta =-\left({\frac {5}{8}}m^{4}+{\frac {30}{8}}m^{2}n^{2}+{\frac {15}{8}}n^{4}\right){\mbox{.}}}$
(22)

and

 ${\displaystyle u_{1}={\frac {\left(8\eta n+5n^{5}+30m^{2}n^{3}+15m^{4}n\right)}{8\left({\beta }^{2}-{\Omega }^{2}\right)}}\left(cos\beta t-\right.}$${\displaystyle \left.cos\Omega t\right)-{\frac {\left(5m^{5}+20m^{3}n^{2}\right)}{128{\beta }^{2}}}\left(cos\beta t-\right.}$${\displaystyle \left.cos3\beta t\right)+{\frac {\left(5n^{5}+20m^{2}n^{3}\right)}{16\left({\beta }^{2}-9{\Omega }^{2}\right)}}\left(cos\beta t-\right.}$${\displaystyle \left.cos3\Omega t\right)-{\frac {m^{5}}{384{\beta }^{2}}}\left(cos\beta t-\right.}$${\displaystyle \left.cos5\beta t\right)+{\frac {n^{5}}{16\left({\beta }^{2}-25{\Omega }^{2}\right)}}\left(cos\beta t-\right.}$${\displaystyle \left.cos5\Omega t\right)+{\frac {5m^{4}n}{16\left[{\beta }^{2}-{\left(4\beta +\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(4\beta +\Omega )t\right)+{\frac {20m^{4}n}{16\left[{\beta }^{2}-{\left(2\beta +\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(2\beta +\Omega )t\right)+{\frac {20m^{4}n}{16\left[{\beta }^{2}-{\left(2\beta -\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(2\beta -\Omega )t\right)+{\frac {5m^{4}n}{16\left[{\beta }^{2}-{\left(4\beta -\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(4\beta -\Omega )t\right)+{\frac {5m^{3}n^{2}}{8\left[{\beta }^{2}-{\left(5m3\beta +2\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(3\beta +2\Omega )t\right)+{\frac {15m^{3}n^{2}}{8\left[{\beta }^{2}-{\left(\beta +2\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(\beta +2\Omega )t\right)+{\frac {15m^{3}n^{2}}{8\left[{\beta }^{2}-{\left(\beta -2\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(\beta -2\Omega )t\right)+{\frac {5m^{3}n^{2}}{8\left[{\beta }^{2}-{\left(3\beta -2\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(3\beta -2\Omega )t\right)+{\frac {5m^{2}n^{3}}{8\left[{\beta }^{2}-{\left(2\beta +3\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(2\beta +3\Omega )t\right)+{\frac {15m^{2}n^{3}}{8\left[{\beta }^{2}-{\left(2\beta +\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(2\beta +\Omega )t\right)+{\frac {15m^{2}n^{3}}{8\left[{\beta }^{2}-{\left(2\beta -\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(2\beta -\Omega )t\right)+{\frac {5m^{2}n^{3}}{8\left[{\beta }^{2}-{\left(2\beta -3\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(2\beta -3\Omega )t\right)+{\frac {5m\quad n^{4}}{16\left[{\beta }^{2}-{\left(\beta +4\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(\beta +4\Omega )t\right)+{\frac {20m\quad n^{4}}{16\left[{\beta }^{2}-{\left(\beta +2\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(\beta +2\Omega )t\right)+{\frac {20m\quad n^{4}}{16\left[{\beta }^{2}-{\left(\beta -2\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(\beta -2\Omega )t\right)+{\frac {5m\quad n^{4}}{16\left[{\beta }^{2}-{\left(\beta -4\Omega \right)}^{2}\right]}}\left(cos\beta t-\right.}$${\displaystyle \left.cos(\beta -4\Omega )t\right){\mbox{.}}}$
(23)

Using Eq. (8) the perturbation solution can be expressed as follows:

 ${\displaystyle u(t)=u_{0}+\alpha u_{1}{\mbox{,}}}$
(24)

where ${\textstyle u_{0}}$ and ${\textstyle u_{1}}$ are given by Eqs. (21) and (23). As we did before we check numerically the validity of Eq. (17) by calculating the solution of Eq. (24) and also, a good agreement is found for different values of A and ε as shown in Fig. 2.

 Figure 2. Comparison of the approximate solution (solid line) with the numerical solution (dashed line).

## 4. Conclusion

A new perturbation technique valid for large parameter is employed in the present paper. It is an effective method to determine analytical solution for strongly nonlinear Duffing oscillators. All analytical solutions computed in the present paper are in pretty good agreement with those obtained by the numerical solutions. It is worthy to mention that the present method is an extremely simple and leads to high accuracy of the obtained results.

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