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 [22] and is given by.

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 [19] and Equation (23) in Ref [20]). Hence, the approximate periodic solution can be readily obtained:

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 [19] and Equation (23) in Ref [20])

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 [23]:

where

In Figure 1, we presented the variety of comparisons between approximate and exact solution for Equation (1). For It can be observed that Equation (26) provides excellent approximation to the exact periodic solution in Equation (27).

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 [23], and for approximate values we used Equation (49). In fact the exact periodic solution it can be obtained from the following complicated relation that is given in [23].

where

To compare with the exact period, Figure 2 shows the comparison between the approximate periodic solution obtained from formula (49) and the exact periodic solution obtained from formula (50) see [23]. It can be seen from Figure 2 that the approximate periodic solution is nearly identical with that given by the exact periodic solution.