Approximate solutions of the fractional damped nonlinear oscillator subject to Van der Pol system

Abstract

This paper deals with fractal Van der Pol damped nonlinear oscillators equation having nonlinearity. By combining the techniques of the Laplace transform and the variational iteration method, we establish approximate periodic solutions for the fractal damped nonlinear systems. In this simple way, nonlinear differential equations can be easily converted into linear differential equations. Illustrative examples including the Van der Pol damped nonlinear oscillator reveal that this method is very effective and convenient for solving fractal nonlinear differential equations. Finally, comparison of the obtained results with those of the other achieved method, also reveals that this coupling method not only suggests an easier method due to the Lagrange multiplier but also can be easily extended to other nonlinear systems.

Introduction

The study of nonlinear oscillators has caught much attention in the development of the theory of dynamical systems recently. Especially the Van der Pol oscillator can be described a nonlinear mass-spring-damper system which has been used to develop models in many applications, such as electronics, biology or acoustics. The dynamics of the nonlinear oscillators are of interest for its current and potential applications and to find its approximate analytical solutions. So far there have been many widely used analytical techniques developed for solving nonlinear differential equations in many engineering oscillatory systems. For example, Variational iteration method,^1–6 Homotopy perturbation method,^7–10 Iteration perturbation method,^11–13 Two-scale method,^14–16 Energy balance method^17–19 and Frequency–amplitude formulation.^20–22

To deal with the nonlinear oscillation systems problems, the application of the variational iteration method in nonlinear problems has been developed by scientists and engineers because this method could transform the difficult problem into a simple problem which is easy to solve. The variational iteration method was first proposed by professor He. ²³ The basical idea of this method is to construct a correction functional with a Lagrange multiplier, which is determined by the variational theory.^24–30 Now it has been used to solve many kinds of nonlinear problems. Prof. He.^31,32 ensured the effectiveness of the combining techniques of the Laplace transform and the variational iteration method in Micro-electro-mechanical system. In the solution process, the Laplace transform is suitable for nonlinear oscillators and makes the construction of the Lagrange multiplier much simpler. ³³

In this manuscript, under the two initial conditions, coupled the variational iteration method with the Laplace transform which improves simplicity, flexibility and accuracy is employed to get the approximate solution of the fractional Van der Pol damped nonlinear oscillator. We show that the influence of the parameter of nonlinear frequency and the approximate analytic solutions to this problem.

Numerical approximation method for nonlinear oscillators

The general form of nonlinear oscillator can be represented with second-order nonlinear ordinary differential equation as follows:

$\frac{d^{2}u}{d{t}^2}+f\left(u\right)=0$

(1)

We can re-express equation (1) as

$\frac{d^{2}u}{d{t}^2}+{\omega }^{2}u+g\left(u\right)=0$

(2)

where $g\left(u\right)=f\left(u\right)-{\omega }^{2}u$ and $\omega$ is the frequency of the oscillation of the system to be determined.

According to variational iteration method, we can construct the correction functional for equation (2) as

$u_{k+1}\left(t\right)=u_k\left(t\right)+{{\displaystyle \int }}_0^t\lambda \left(\xi \right)\left[\frac{d^{2}u\left(\xi \right)}{d{\xi }^2}+{\omega }^2{u}_k\left(\xi \right)+{\stackrel{\sim }{g}}_k\left(\xi \right)\right]\hspace{0.33em}d\xi ,k=0,\hspace{0.33em}1,\hspace{0.33em}2,\hspace{0.33em}\dots$

(3)

where $\lambda$ is a general Lagrange multiplier. Recently, some authors adopt the general form of Lagrange multiplier as

$\lambda =\overline{\lambda }\left(t-\xi \right)$

Applying the Laplace transform on both side of equation (3), the correction functional can be written as

$L\left[u_{k+1}\left(t\right)\right]=L\left[u_k\left(t\right)\right]+L\left[{{\displaystyle \int }}_0^t\lambda \left(t-\xi \right)\left[\frac{d^2{u}_k\left(\xi \right)}{d{\xi }^2}+{\omega }^2{u}_k\left(\xi \right)+{\stackrel{\sim }{g}}_k\left(u\right)\right]d\xi \right]$

(4)

Thus,

$L\left[u_{k+1}\left(t\right)\right]=$

$L\left[u_k\left(t\right)\right]+L\left[\lambda \left(t\right)\right]L\left[\left(s^2+{\omega }^2\right)L\left[u_k\left(t\right)\right]-s{u}_k\left(0\right)-{\frac{d{u}_k}{dt}|}_{t=0}+L\left[{\stackrel{\sim }{g}}_k\left(u\right)\right]\right]$

(5)

Here, $k$ represents the $k$ th approximation and ${\stackrel{\sim }{g}}_k$ is a restricted variation an $\delta {\stackrel{\sim }{g}}_k=0$ . We get

$L\left[\delta u_{k+1}\right]=L\left[\delta u_k\right]+L\left[\lambda \left(t\right)\right]\left(s^2+{\omega }^2\right)L\left[\delta u_k\left(t\right)\right]$

(6)

$L\left[\lambda \left(t\right)\right]=-\frac{1}{\left(s^2+{\omega }^2\right)}$

(7)

By applying the inverse Laplace transform on the equation (7), the Lagrange multiplier

$\lambda \left(t\right)=-\frac{1}{\omega }\mathit{sin}\omega t$

(8)

Using equation (4), the form of the iterative formula as follows $L\left[u_{k+1}\left(t\right)\right]=L\left[u_k\left(t\right)\right]-\frac{1}{\omega }L\left[\underset{0}{\overset{t}{{\displaystyle \int }}}\mathit{sin}\omega \left(t-\xi \right)\left[\frac{d^2{u}_k\left(\xi \right)}{d{\xi }^2}+{\omega }^2{u}_k\left(\xi \right)+{\stackrel{\sim }{g}}_k\left(u\right)\right]d\xi \right]$

After some simple calculations, we get the correction functional is further reduced with the initial conditions $u\left(0\right)=\frac{\text{du}\left(0\right)}{dt}=0$

$L\left[u_{k+1}\left(t\right)\right]=-\frac{1}{\omega }L\left[{{\displaystyle \int }}_0^t\mathit{sin}\omega \left(t-\xi \right)\left[{\stackrel{\sim }{g}}_k\left(u\right)\right]d\xi \right]$

The iterative formula for equation (1) after the convolution operation with the zero initial conditions

$L\left[u_{k+1}\left(t\right)\right]=-\frac{1}{\omega }L\left[\mathit{sin}\omega t\right]\hspace{0.33em}L\left[g\left(u_k\left(t\right)\right)\right]$

(9)

Application

A modified fractional Van der Pol Oscillator

We consider the following modified fractional Van der Pol damped nonlinear oscillator can be represented by the following nonlinear equation^34–38

$\frac{d^{2}u}{d{t}^2}+u^{\frac{1}{3}}+\epsilon \left(1-u^2\right)\frac{du}{dt}=0$

(10)

here, $\epsilon$ is a small positive parameter with initial conditions

$u\left(0\right)=A,\frac{\text{du}\left(0\right)}{\text{dt}}=0$

(11)

By Laplace based variational iteration method discussed in the above section and according to equation (2), equation (10) can be rewrite as

$\frac{d^{2}u}{d{t}^2}+{\omega }^{2}u+g\left(u\right)=0$

(12)

where $g\left(u\right)=-{\omega }^{2}u+\epsilon \left(1-u^2\right)-u^{\frac{1}{3}}$

The iteration formula for equation (12) using equation (9) as

$\begin{array}{ccc}L\left[u_{k+1}\left(t\right)\right]& =& L\left[u_k\left(t\right)\right]-L\left[{{\displaystyle \int }}_0^{\infty }\frac{1}{w}\mathit{sin}\omega \left(t-\xi \right)\left[u_k^{″}\left(\xi \right)+{\omega }^2{u}_k\left(\xi \right)+g\left(u_k\right)\right]d\xi \right]\\ & & =L\left[u_k\left(t\right)\right]-\frac{1}{w}L\left[\mathit{sin}\omega t\right]L\left[u_k^{″}\left(\xi \right)+{\omega }^2{u}_k\left(\xi \right)+g\left(u_k\right)\right]=L\left[u_k\left(t\right)\right]-\frac{1}{w}L\left[\mathit{sin}\omega t\right]L\left[u^{″}+u^{\frac{1}{3}}+\epsilon \left(1-u^2\right)u^{\prime }\right]\end{array}$

(13)

Solving equation (11), we have

$u_0\left(t\right)=A\mathit{cos}\omega t$

Substituting $u_0\left(t\right)$ into equation (13), after a simple calculation, we have

$\begin{array}{ccc}L\left[u_1\left(t\right)\right]& =& L\left[A\mathit{cos}\omega t\right]-\frac{1}{\omega }L\left[\mathit{sin}\omega t\right]L\left[{\left(A\mathit{cos}\omega t\right)}^{\frac{1}{3}}+\epsilon (1-A^2{\mathit{cos}}^2\omega t)A\omega \mathit{sin}\omega t-{\omega }^{2}A\mathit{cos}\omega t\right]\\ & & L\left[A\mathit{cos}\omega t\right]-\frac{1}{\omega }L\left[\mathit{sin}\omega t\right]L[\epsilon A^3\omega \left(\mathit{sin}\omega t-{\mathit{sin}}^3\omega t\right)+\frac{\epsilon A^3\omega }{4}\mathit{sin}\omega t\\ & & +a_1{A}^{\frac{1}{3}}\mathit{cos}\omega t-a_2{A}^{\frac{1}{3}}\mathit{cos}3\omega t-\epsilon A\omega \mathit{sin}\omega t-{\omega }^{2}A\mathit{cos}\omega t]\\ & & \left[A\mathit{cos}\omega t\right]-\frac{1}{\omega }L\left[\mathit{sin}\omega t\right]L\left[\left(A^{\frac{1}{3}}{a}_1-{\omega }^{2}A\right)\mathit{cos}\omega t\right]-\frac{1}{\omega }L\left[\mathit{sin}\omega t\right]L\left[\frac{\epsilon A^3\omega }{4}\mathit{sin}3\omega t\right]-\frac{1}{\omega }L\left[\mathit{sin}\omega t\right]L\\ & & \left[A^{\frac{1}{3}}{a}_2\mathit{cos}3\omega t\right]-\frac{1}{\omega }L\left[\mathit{sin}\omega t\right]L\left[\left(\frac{\epsilon A^3\omega }{4}-\epsilon A\omega \right)\mathit{sin}\omega t\right]\end{array}$

(14)

Here, the Fourier series ${\left(A\cos \omega t\right)}^{\frac{1}{3}}=a_1\mathit{cos}\omega t+b\mathit{cos}3\omega t+\cdots$ has been calculated ³⁹

Then no secular terms requires that

$4a_1{A}^{\frac{1}{3}}-4A{\omega }^2=0$

From the above equation, we can easily get

$A=2,\omega =0.8574$

which is same with that obtained in Ref [40].

So the approximate solution can be obtained

$u_{app}\left(t\right)=2\mathit{cos}\left(0.8574t\right)$

A modified Fractional Van der Pol Oscillator with two-point boundary conditions

we consider the equation (10) with the two-point boundary conditions

$u\left(0\right)=A,u\left(\frac{T}{4}\right)=0$

(15)

The iteration formula for equation (10)

$\begin{array}{ccc}{u}_{k+1}\left(t\right)& =& u_k\left(t\right)+\frac{\mathit{cos}\omega t}{\omega }{{\displaystyle \int }}_0^t\mathit{sin}\omega \xi \left[u^{″}\left(\xi \right)+{u_k}^{\frac{1}{3}}\left(\xi \right)+\epsilon \left(1-{u_k}^2\left(\xi \right)\right){u_k}^{\prime }\left(\xi \right)\right]d\xi \\ & & +\frac{\mathit{sin}\omega t}{\omega }{{\displaystyle \int }}_t^{\frac{T}{4}}\mathit{cos}\omega \xi \left[u^{″}\left(\xi \right)+{u_k}^{\frac{1}{3}}\left(\xi \right)+\epsilon \left(1-{u_k}^2\left(\xi \right)\right){u_k}^{\prime }\left(\xi \right)\right]d\xi \end{array}$

(16)

Due to the initial conditions in equation (11)

$u_0\left(t\right)=A\mathit{cos}\omega t$

Substituting $u_0\left(t\right)$ into equation (16), we get

$\begin{array}{ccc}{u}_1\left(t\right)& =& u_0\left(t\right)+\frac{\mathit{cos}\omega t}{\omega }{{\displaystyle \int }}_0^t\mathit{sin}\omega \xi \left[u^{″}\left(\xi \right)+{u_0}^{\frac{1}{3}}\left(\xi \right)+\epsilon \left(1-{u_0}^2\left(\xi \right)\right){u_0}^{\prime }\left(\xi \right)\right]d\xi \\ & & +\frac{\mathit{sin}\omega t}{\omega }{{\displaystyle \int }}_t^{\frac{T}{4}}\mathit{cos}\omega \xi \left[u^{″}\left(\xi \right)+{u_0}^{\frac{1}{3}}\left(\xi \right)+\epsilon \left(1-{u_0}^2\left(\xi \right)\right){u_0}^{\prime }\left(\xi \right)\right]d\xi \end{array}$

(17)

After a simple calculation, we have

$\begin{array}{ccc}{u}_1\left(t\right)& =& A\mathit{cos}\omega t+\frac{\mathit{cos}\omega t}{\omega }{{\displaystyle \int }}_0^t\mathit{sin}\omega \xi \left[{\left(A\mathit{cos}\omega t\right)}^{\frac{1}{3}}+\epsilon (1-A^2{\mathit{cos}}^2\omega t)A\omega \mathit{sin}\omega t-{\omega }^{2}A\mathit{cos}\omega t\right]d\xi \\ & & +\frac{\mathit{sin}\omega t}{\omega }{{\displaystyle \int }}_t^{\frac{T}{4}}\mathit{cos}\omega \xi \left[{\left(A\mathit{cos}\omega t\right)}^{\frac{1}{3}}+\epsilon (1-A^2{\mathit{cos}}^2\omega t)A\omega \mathit{sin}\omega t-{\omega }^{2}A\mathit{cos}\omega t\right]d\xi \\ & & =A\mathit{cos}\omega t+\frac{\mathit{cos}\omega t}{\omega }{{\displaystyle \int }}_0^t\mathit{sin}\omega \xi [\left(A^{\frac{1}{3}}{a}_1-{\omega }^{2}A\right)\mathit{cos}\omega t-A^{\frac{1}{3}}{a}_2\mathit{cos}3\omega t-\left(\frac{\epsilon A^3\omega }{4}-\epsilon A\omega \right)\mathit{sin}\omega t\\ & & -\frac{\epsilon A^3\omega }{4}\mathit{sin}3\omega t]d\xi +\frac{\mathit{sin}\omega t}{\omega }{{\displaystyle \int }}_t^{\frac{T}{4}}\mathit{cos}\omega \xi [\left(A^{\frac{1}{3}}{a}_1-{\omega }^{2}A\right)\mathit{cos}\omega t-A^{\frac{1}{3}}{a}_2\mathit{cos}3\omega t\\ & & -\left(\frac{\epsilon A^3\omega }{4}-\epsilon A\omega \right)\mathit{sin}\omega t-\frac{\epsilon A^3\omega }{4}\mathit{sin}3\omega t]d\xi \hspace{1em}\end{array}$

(18)

Here,

$4a_1{A}^{\frac{1}{3}}-4A{\omega }^2=0$

We can also get

$A=2,\omega =0.8574$

The approximate solution

$u_{app}\left(t\right)=2\mathit{cos}\left(0.8574t\right)$

Conclusions

In summary, we have demonstrated the applicability of the variational iteration method with the Laplace transform to solve nonlinear fractional Van der Pol damped nonlinear oscillators with two boundary conditions. The method which we applied is extremely simple, easy to find the approximate solution of the fractional nonlinear oscillator. To our knowledge, this study showed that the coupling method was very effective way for solving nonlinear differential equations which can also be easily extended to other nonlinear boundary value problem and has a great potential in many areas of physics and engineering which still needs further development.