Homotopy perturbation method coupled with the enhanced perturbation method

Abstract

The enhanced perturbation method is used to improve a governing equation to a higher order, followed by the classic perturbation method. This paper adopts the basic idea of the method to construct a homotopy equation with a higher order. The results show that this modification of the homotopy perturbation method leads to a very high accuracy of the obtained solution. Its solution process is elucidated by using Duffing oscillator as an example, and the obtained frequency is valid for all amplitudes from an extremely small value to infinity with a maximal error of 3.34%. The main merit of this higher order homotopy perturbation method is that the obtained frequency is valid for the whole solution domain.

Introduction

Homotopy perturbation method is a coupled method of the classical perturbation method and the homotopy technology, and it can be widely used to solve various non-linear problems.^1–14 The development of the perturbation method can boost the homotopy perturbation method. The multiple scales in the perturbation method were adopted by El-Dib, and the multiple scales homotopy perturbation method can effectively solve various forced non-linear vibrations. ¹⁵ Recently, Filobello-Nino et al. proposed a modification of perturbation method, called the enhanced perturbation method. ¹⁶ This modification overcomes all shortcomings of the perturbation method. To give a brief introduction to the enhanced perturbation method, we consider the following Duffing equation

$$u^{″}+u+\epsilon u^3=\cos \beta t$$

(1)

Equation (1) can be written in the form

$$\left(D^2+1+\epsilon u^2\right)u=\sin \beta t$$

(2)where D is a simple differential operator D = d/dt.

According to the enhanced perturbation method, ¹⁶ we can apply the so-called annihilator operator $D^2+\beta$ to the above equation, and this results in

$$\left(D^2+\beta \right)\left(D^2+1+\epsilon u^2\right)u=0$$

(3)

The enhanced perturbation method can solve wide classes of non-linear problems, where the classic perturbation method fails. ¹⁶

A modified homotopy perturbation method in view of the enhanced perturbation method

The enhanced perturbation method is extremely effective for forced oscillators. ¹⁶ In this paper, we consider the Duffing equation without a forced term

$$u^{″}+u+\epsilon u^3=0$$

(4)with initial conditions

$$u\left(0\right)=A$$

(5)

$$u^{\prime }\left(0\right)=0$$

(6)

To illustrate the basic idea to couple the enhanced perturbation method in the solution process of the homotopy perturbation method, a linear oscillator is considered as follows

$$u^{″}+{\omega }^{2}u=\left(D^2+{\omega }^2\right)u=0$$

(7)

The enhanced perturbation method ¹⁶ leads to the following higher order differential equation

$$\left(D^2+{\omega }^2\right)\left(D^2+{\omega }^2\right)u=u^{″″}+2{\omega }^2{u}^{″}+{\omega }^{4}u=0$$

(8)

Replacing $u^{″}$ by $-{\omega }^{2}u$ in equation (7), we have

$$u^{″″}-{\omega }^{4}u=0$$

(9)

We give a similar way to improve the order of equation (4). Differentiating equation (4) with respect to time twice, we have

$$u^{‴}+u^{\prime }+3\epsilon u^2{u}^{\prime }=0$$

(10)

$$u^{″″}+u^{″}+6\epsilon u{u^{\prime }}^2+3\epsilon u^2{u}^{″}=0$$

(11)

Replacing $u^{″}$ by $-\left(u+\epsilon u^3\right)$ , we have

$$u^{″″}-\left(1+3\epsilon u^2\right)\left(u+\epsilon u^3\right)+6\epsilon u{u^{\prime }}^2=0$$

(12)or

$$u^{″″}-u-4\epsilon u^3-3{\epsilon }^2{u}^5+6\epsilon u{u^{\prime }}^2=0$$

(13)

A simple homotopy equation can be established

$$u^{″″}-1\times u+p\left(-4\epsilon u^3-3{\epsilon }^2{u}^5+6\epsilon u{u^{\prime }}^2\right)=0$$

(14)

The solution is expanded into a series of p

$$u=u_0+p{u}_1+p^2{u}_2+\cdots$$

(15)

Using the parameter expansion technology,^6,17 and the coefficient of the linear term, 1, is also expanded into a series of p

$$1={\omega }^4+p{\omega }_1+p^2{\omega }_2+\cdots$$

(16)

Substituting equations (15) and (16) into equation (14), expanding the resultant equation and collecting the same powers of p, proceeding the same way as the perturbation method, we have

$${u^{″″}}_0-{\omega }^4{u}_0=0$$

(17)

$${u^{″″}}_1-{\omega }^4{u}_1-{\omega }_1{u}_0-4\epsilon u_0^3-3{\epsilon }^2{u}_0^5+6\epsilon u_0{u^{\prime }}_0^2=0$$

(18)

By the initial conditions, equations (5) and (6), we can solve $u_0$ , which reads

$$u_0=A\cos \omega t$$

(19)

Substituting $u_0$ into equation (18), we have

$${u^{″″}}_1-{\omega }^4{u}_1-{\omega }_{1}A\cos \omega t-4{\epsilon }_A^3{\cos }^3\omega t-3{\epsilon }^2{A}^5{\cos }^5\omega t+6\epsilon A^3{\omega }^2\cos \omega t{\sin }^2\omega t=0$$

(20)

By a simple manipulation, equation (20) can be re-written in the form

$$\begin{array}{c}{u^{″″}}_1-{\omega }^4{u}_1-{\omega }_{1}A\cos \omega t-4{\epsilon }_A^3\left(\frac{3}{4}\cos \omega t+\frac{1}{4}\cos 3\omega t\right)\\ -3{\epsilon }^2{A}^5\left(\frac{10}{16}\cos \omega t+\frac{5}{16}\cos 3\omega t+\frac{1}{16}\cos 5\omega t\right)\\ +6\epsilon A^3{\omega }^2\left(\frac{1}{4}\cos \omega t-\frac{1}{4}\cos 3\omega t\right)=0\end{array}$$

(21)or

$$\begin{array}{c}{u^{″″}}_1-{\omega }^4{u}_1+\left(-{\omega }_{1}A-3{\epsilon }_A^3-\frac{30}{16}{\epsilon }^2{A}^5+\frac{3}{2}\epsilon A^3{\omega }^2\right)\cos \omega t\\ +\left(-{\epsilon }_A^3-\frac{15}{16}{\epsilon }^2{A}^5-\frac{3}{2}\epsilon A^3{\omega }^2\right)\cos 3\omega t-\frac{3}{16}{\epsilon }^2{A}^5\cos 5\omega t=0\end{array}$$

(22)

The solution of u₁ should avoid $t\cos \omega t$ or $t\sin \omega t$ , which is called as secular terms in perturbation theory, and this requires

$$-{\omega }_{1}A-3{\epsilon }_A^3-\frac{30}{16}{\epsilon }^2{A}^5+\frac{3}{2}\epsilon A^3{\omega }^2=0$$

(23)

If we search for only the first-order approximate solution, setting p = 1 in equation (16), we have

$$1={\omega }^4-3{\epsilon }_A^2-\frac{15}{8}{\epsilon }^2{A}^4+\frac{3}{2}\epsilon A^2{\omega }^2$$

(24)

Solving ${\omega }^2$ from equations (23) and (24), we have

$$\begin{array}{c}{\omega }^2=\frac{-\frac{3}{2}\epsilon A^2+\sqrt{\frac{9}{4}{\epsilon }^2{A}^4+4\left(1+3{\epsilon }_A^2+\frac{15}{8}{\epsilon }^2{A}^4\right)}}{2}\\ =\frac{-\frac{3}{2}\epsilon A^2+\sqrt{4+12{\epsilon }_A^2+9.75{\epsilon }^2{A}^4}}{2}\end{array}$$

(25)

In order to verify the accuracy of our result, we consider two extreme cases: (1) When $\epsilon \to 0$ or $A\to 0$ , equation (25) can be simplified as

$${\omega }^2=\frac{-\frac{3}{2}\epsilon A^2+4\left(1+\frac{12}{8}{\epsilon }_A^2\right)}{2}=1+\frac{3}{4}\epsilon A^2$$

(26)

This is exactly same as that obtained by the classic homotopy perturbation method^6,11 (2) when $\epsilon \to \infty$ or $A\to \infty$ equation (25) becomes

$${\omega }^2=\frac{-\frac{3}{2}\epsilon A^2+\sqrt{9.75{\epsilon }^2{A}^4}}{2}=\frac{1}{2}\left(-\frac{3}{2}+\sqrt{9.75}\right)\epsilon A^2=0.8112\epsilon A^2$$

(27)or

$$\omega =0.9007{\epsilon }^{1/2}A$$

(28)the exact frequency when $\epsilon A^2\to \infty$ is^6,11

$${\omega }_{ex}=0.9318\sqrt{\epsilon A^2}$$

(29)

That means our result is valid for all $0<\epsilon A^2<\infty$ with a maximal relative error of 3.34%, much better than that by the classic homotopy perturbation method.

Conclusions

The enhanced perturbation method has many fascinating properties; it can easily deal with non-linear differential equations by improving the order of the original differential equation. ¹⁶ This paper shows the basic idea of the enhanced perturbation method can be adopted in the homotopy perturbation method, making the solution process much more reliable and accurate. We recommend this modification of the homotopy perturbation method as the higher order homotopy perturbation method, and the results are valid for all $0<\epsilon A^2<\infty$ , with a maximal error of 3.34%.