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In this study, homotopy perturbation method and parameter expanding method are applied to the motion equations of two nonlinear oscillators. Our results show that both the (HPM) and (PEM) yield the same results for the nonlinear problems. In comparison with the exact solution, the results show that these methods are very convenient for solving nonlinear equations and also can be used for strong nonlinear oscillators.

The study of nonlinear oscillators has been important in the development of the theory of dynamical systems. The Van der Pol oscillator can be regarded as describing a mass-spring-damper system with a nonlinear positiondependent damping coefficient or, equivalently, an RLC electrical circuit with a negative nonlinear resistor, and has been used to develop models in many applications, such as electronics, biology or acoustics. It represents a nonlinear system with an interesting behavior that arises naturally in several applications. Very recently, various kinds of analytical and numerical methods have been used to solve the problems of nonlinear oscillators, such as Frequency-amplitude formulation [1-3], Energy balance method [4-6], Variational iteration method [7,8], Homotopy perturbation method [9-12] and Parameter expanding method [13-15].

In recent years, the application of the homotopy perturbation method in nonlinear problems has been developed by scientists and engineers, because this method continuously deforms the difficult problem under study into a simple problem which is easy to solve. The homotopy perturbation method was proposed first by He in 1998 and was further developed and improved by him [10,13,16]. Homotopy is an important part of differential topology. The homotopy perturbation method is, in fact, a coupling of the traditional perturbation method and homotopy in topology [

Parameter expanding method proposed by He [

This paper applies (HPM) and (PEM) to fractional Van der Pol damped nonlinear oscillator. Comparison of the period of oscillation and the exact solution shows that both methods are very effective and convenient and quite accurate to nonlinear engineering problems.

In order to show the accuracy of homotopy perturbation method (HPM) and parameter expanding method (PEM) for solving nonlinear equations and to compare it with exact solutions, we will consider the following examples.

The classical fractional Van der Pol damped nonlinear oscillator can be represented by the following nonlinear equation [19-21].

with the initial conditions:

The following homotopy can be constructed

As in He’s homotopy perturbation method [9,10,16], it is obvious that when p = 0 Equation (3) becomes the linearized equation, for p = 1 Equation (3) then becomes the original problem. Assume that the periodic solution to Equation (3) may be written as a power series in p:

Setting p = 1, leads to the approximate solution of the problem:

Substituting Equation (4) into Equation (3) and equating the terms with the identical powers of p,

Solving Equation (6), we have

The Fourier series for has been calculated [

where,

Substituting Equation (8) into Equation (7) leads to

Eliminating the secular term, we have

From the above equation, we can easily find that

which is same with that obtained by iteration procedure (see Equation (40) in Ref [

We rewrite Equation (1) in the form

To solve Equation (1) by parameter expanding method we expand the solution of the problem and coefficients 0 and 1 in the left side of Equation (14) in series of as follows

Substituting Equations (15)-(17) into Equation (14), and processing as the standard perturbation method, we have

The solution of Equation (18) can be easily obtained

Substituting the result into Equation (19) yields

Using the relation (9) Equation (21) can be rewritten as

No secular terms requires

If the first order approximation is enough, then setting in Equations (15)-(17) yields

From relation (23) and (24) we have

which is the same with that obtained by iteration procedure in [19,20] (see Equation (40) in Ref [

Hence, the approximate periodic solution can be readily obtained:

To illustrate and verify the accuracy of this method, we may compare the approximate periodic solution and the exact periodic solution. For reference, the exact solution for the classical fractional Van der Pol damped nonlinear oscillator is as follows [

where

In

The special case of the fractional Van der Pol damped nonlinear oscillator or the Rayleigh equation can be represented by [21,24,25].

with the initial conditions:

We can establish the following homotopy

Similarly, when p = 0, Equation (31) becomes a linear equation; for p = 1, Equation (31) then becomes the original problem. Applying the perturbation technique, the solution of Equation (31) can be expressed as a power series in p:

Setting p = 1, leads to the approximate solution of the problem:

Substituting Equation (32) into Equation (31) and equating the terms with the identical powers of p,

The solution for is

Substituting Equation (36) into Equation (35) leads to

No secular terms requires

From the above equation, we can easily find that

Hence, the approximate periodic solution can be readily obtained:

We rewrite Equation (29) in the form

Substituting Equations (15)-(17) into Equation (41), collecting the same power of and equating each coefficient of to zero, we obtain

Solving Equation (42), we have

Substituting into Equation (43) results in

Eliminating the secular terms needs

If the first order approximation is enough, then setting in Equations (15)-(17) yields

From relation (46) and (47) we have

Hence, the approximate periodic solution can be readily obtained:

To write the exact periodic solution we used its values from [

where

To compare with the exact period,

In this paper, we give a comparative study between Homotopy perturbation method (HPM) and Parameter expanding method (PEM) to obtain the approximate periodic solutions to fractional Van der Pol damped nonlinear oscillators. Illustrative examples reveal that these methods are very effective and convenient for solving nonlinear differential equations. Comparisons are also made between the exact solution and the results of the Homotopy perturbation method and Parameter expanding method in order to prove the precision of the results obtained from both methods mentioned.