An effective modification of Ji-Huan He’s variational approach to nonlinear singular oscillator

Abstract

In the review article “Asymptotic methods for solitary solutions and compactons” (Ji-Huan He, Abstract and Applied Analysis 2012: 916793), the variational approach and the Hamiltonian approach to nonlinear oscillators are systematically discussed. This paper gives an extension of the methods to singular nonlinear oscillator by the iteration perturbation method.

Introduction

Chinese mathematician, Prof. Ji-Huan He, systematically discussed the variational approach and the Hamiltonian approach to nonlinear oscillators in his review article, ¹ where the history and main development of both methods were elucidated. Although He’s methods ¹ have been proved to be effective to nonlinear oscillators, extension of the methods still exists. Consider a nonlinear oscillator in the form

$$\ddot{x}+f\left(x\right)=0$$

(1)

with initial conditions

$$x\left(0\right)=A\text{ and }\dot{x}\left(0\right)=0$$

(2)

where a dot denotes the differentiation with respect to time $t$ , $A$ is a given constant, $-f(x)$ is a conservative force.

There are many approaches to nonlinear oscillators, for example, the homotopy perturbation method,^2–5 He’s frequency–amplitude formulation^6–8 and other frequency–amplitude formulae,^9–14 Akbari-Ganji method,^15,16 and reviews on various methods are available in the literature.^17–19 In this paper, we will give an extension of He’s variational approach and Hamiltonian approach. ¹

A variational principle for equation (1) can be easily established, which reads^20–24

$$J\left(x\right)={{\displaystyle \int }}^{\text{}}\left\{\frac{1}{2}{\dot{x}}^2-F\left(x\right)\right\}\text{d}t$$

(3)

where F is the potential, defined as $\partial F$ / $\partial$ x = f.

The variational approach and the Hamiltonian approach discussed by He ¹ can be applied to equation (3) to obtain the frequency of the nonlinear oscillator. This comment shows that we can couple the iteration perturbation method ²⁵ with the variational approach or the Hamiltonian approach to search for a periodic solution.

Basic process

The oscillator (1)–(2) can be rewritten in a system form

$$y=\dot{x},\dot{y}=-f\left(x\right),x\left(0\right)=A,y\left(0\right)=0$$

(4)

Introducing polar representations for $\left(x,y\right)$ , i.e.

$$x\left(t\right)=r\left(t\right)\cos \hspace{0.17em}\hspace{0.17em}\theta \left(t\right),y\left(t\right)=r\left(t\right)\sin \hspace{0.17em}\hspace{0.17em}\theta \left(t\right)$$

(5)

with initial conditions

$$r\left(0\right)=A\text{ and }\theta \left(0\right)=0$$

(6)

Substituting equation (5) into equation (4), we have

$$\dot{r}\cos \hspace{0.17em}\theta =\left(1+\dot{\theta }\right)r\sin \hspace{0.17em}\theta$$

(7)

$$\dot{r}\sin \hspace{0.17em}\theta +r\dot{\theta }\cos \hspace{0.17em}\hspace{0.17em}\theta =-f\left(r\cos \hspace{0.17em}\theta \right)$$

(8)

It is easy to arrive at the following results

$$\dot{\theta }=-1+{\cos }^2\theta -\frac{\cos \hspace{0.17em}\theta \cdot f\left(r\cos \hspace{0.17em}\theta \right)}{r}$$

(9)

$$\dot{r}=\frac{r\sin \hspace{0.17em}\left(2\theta \right)}{2}-\sin \hspace{0.17em}\hspace{0.17em}\theta \cdot f\left(r\cos \hspace{0.17em}\theta \right)$$

(10)

According to the iteration perturbation method, ²⁵ we can construct the following iteration scheme

$${\dot{r}}_{k+1}\left(t\right)=F\left[r_k\left(t\right),{\theta }_k\left(t\right)\right],{\dot{\theta }}_{k+1}\left(t\right)=G\left[r_k\left(t\right),{\theta }_k\left(t\right)\right]$$

(11)

where $k=0,1,2\dots ,$ and $F\left(r,\theta \right)$ and $G\left(r,\theta \right)$ are, respectively, the right-hand sides of equations (9) and (10) based on having prior functional forms for $r_0\left(t\right)$ and ${\theta }_0\left(t\right)$ . By averaging over θ and some simple operations, we can first obtain functions ${\dot{r}}_0\left(t\right)$ and ${\dot{\theta }}_0\left(t\right)$ , and then $r_0\left(t\right)$ and ${\theta }_0\left(t\right)$ . Therefore, approximation to the solution $x\left(t\right)=r_1\left(t\right)\cdot \cos \hspace{0.17em}{\theta }_1\left(t\right)$ can be obtained.

Example

An important and interesting nonlinear differential equation occurs in the modeling of certain phenomena in plasma physics.^26–29 This example corresponds to the odd nonlinear singular oscillator with ε a positive constant

$$\ddot{x}=\frac{-1}{\epsilon x},x\left(0\right)=A,\dot{x}\left(0\right)=0$$

(12)

The variational principle for equation (12) reads^20–24

$$J\left(x\right)={{\displaystyle \int }}^{\text{}}\left\{\frac{1}{2}{\dot{x}}^2-\frac{1}{\epsilon }\ln x\right\}\text{d}t$$

(13)

The corresponding equation system is

$$y=\dot{x},\dot{y}=\frac{-1}{\epsilon x},x\left(0\right)=A,y\left(0\right)=0$$

(14)

With the polar expressions (5) and (6), we have

$$\dot{\theta }=-1+\frac{\epsilon r^2{\cos }^2\theta -1}{\epsilon r^2}$$

(15)

$$\dot{r}=\frac{\left(\epsilon r^2{\cos \hspace{0.17em}}^2\theta -1\right)\sin \hspace{0.17em}\theta }{\epsilon r\cos \hspace{0.17em}\theta }$$

(16)

Integral averaging the right-hand side of equations (15) and (16) over θ with one period, between 0 and 2π, we have

$${\dot{r}}_0\left(t\right)=0,{\dot{\theta }}_0\left(t\right)=-\frac{1}{2}-1/\left(\epsilon r_0^2\right)$$

(17)

Combing with the initial conditions in equation (6) yields

$$r_0\left(t\right)=A,{\theta }_0\left(t\right)=-\frac{t}{2}-t/\left(\epsilon A^2\right)$$

(18)

The iteration formula with $k=0,1,2,\dots$ based on equation (18) becomes

$${\dot{\theta }}_{k+1}\left(t\right)=-1+\frac{\epsilon r_k^2{\cos }^2{\theta }_k-1}{\epsilon r_k^2}$$

(19)

$${\dot{r}}_{k+1}\left(t\right)=\frac{\left(\epsilon r_k^2{\cos \hspace{0.17em}}^2{\theta }_k-1\right)\sin \hspace{0.17em}{\theta }_k}{\epsilon r_k\cos \hspace{0.17em}{\theta }_k}$$

(20)

when $k=0$ ,

$${\dot{\theta }}_1\left(t\right)=-1+\frac{\epsilon r_0^2{\cos }^2{\theta }_0-1}{\epsilon r_0^2}$$

(21)

$${\dot{r}}_1\left(t\right)=\frac{\left(\epsilon r_0^2{\cos }^2{\theta }_0-1\right)\sin \hspace{0.17em}{\theta }_0}{\epsilon r_0\cos \hspace{0.17em}{\theta }_0}$$

(22)

Approximation to the solution $x\left(t\right)=r_1\left(t\right)\cdot \cos \hspace{0.17em}{\theta }_1\left(t\right)$ can be obtained just after finding $r_1\left(t\right)$ and ${\theta }_1\left(t\right)$ from equations (21) and (22) based on equation (18)

$${\theta }_1\left(t\right)=\frac{2-3\epsilon A^2}{2\epsilon A^2}t-\frac{\sin \hspace{0.17em}\left(2Wt\right)}{2\left(2+\epsilon A^2\right)}$$

(23)

$$r_1\left(t\right)=A-\frac{A}{2W}+\frac{1}{W}\left(\frac{A}{2}{\cos }^2\left(Wt\right)-\frac{1}{\epsilon A}\ln \cos \hspace{0.17em}\left(Wt\right)\right)$$

(24)

with the amplitude-dependent function, $W\left(A\right)$ , is

$$W=\frac{2+\epsilon A^2}{2\epsilon A^2}$$

(25)

Conclusions

Usually, frequency, period or frequency–amplitude relationship are the main factors for nonlinear oscillators. He ¹⁸ gave an elementary introduction to nonlinear vibration systems. This paper gives for the first time an alternative approach to singular oscillators, and the results are much better than those in literature.

Hereby, we adopt the iteration method ²⁵ into the variational approach under the polar coordinate system, the modification is also an effective way for more oscillators, and it is extremely effective for singular oscillators as illustrated in this paper.