Iterative Homotopy Harmonic Balance Approach for Determining the Periodic Solution of a Strongly Nonlinear Oscillator

A novel approach about iterative homotopy harmonic balancing is presented to determine the periodic solution for a strongly nonlinear oscillator. This approach does not depend upon the small/large parameter assumption and incorporates the salient features of both methods of the parameter-expansion and the harmonic balance. Importantly, in obtaining the higher-order analytical approximation, all the residual errors are considered in the process of every order approximation to improve the accuracy. With this procedure, the higher-order approximate frequency and corresponding periodic solution can be obtained easily. Comparison of the obtained results with those of the exact solutions shows the high accuracy, simplicity, and efficiency of the approach. The approach can be extended to other nonlinear oscillators in engineering and physics.


Introduction
Considerable attention has been paid to the study of nonlinear problems not only in all areas of engineering but also in physics and other disciplines, since world is nonlinear essentially and most phenomena are modeled by nonlinear differential equations.However, in general, it is very difficult to get exact solutions for those problems.Thus, to find approximate analytic solutions to these nonlinear problems has been the desire of many researchers for a long time.
Among these analytical methods, the iterative homotopy harmonic balancing [21] (denoted as IHHB for abbreviation) is a novel and effective technique for solving some strongly nonlinear oscillators.This approach incorporates the salient features of both methods of the parameter-expansion and the harmonic balance.By constructing a parameter  which is considered as a "small parameter" in the harmonic balance process, the IHHB does not depend upon the large/small parameter assumption and can get high-order analytical approximations easily.Hence, the application of IHHB can be found in various nonlinear problems [21][22][23].
In this paper, we consider the following strongly nonlinear oscillator with nonpolynomial term [6] where the over-dot denotes the derivative with respect to the time  and  is the amplitude of the oscillation.

Mathematical Problems in Engineering
To get higher-order analytical and accurate approximations for the frequency and solution of this oscillator, IHHB solving is set up.More importantly, in obtaining the higherorder analytical approximation, all the residual errors are considered in the process of every order approximation to improve the accuracy.Excellent agreement of the approximate solution and frequency with the exact ones has been demonstrated and discussed.As can be seen, the results presented in this paper reveal that IHHB is effective and convenient for some nonlinear oscillators with nonpolynomial terms.

Basic Ideas of Iterative Homotopy Harmonic Balancing Approach
Consider the following second-order systems: where the over-dot denotes the derivative with respect to the time  and  is the amplitude of the oscillations.For convenience, we assume (−) = −().
By introducing a new independent variable  = , then (2) becomes where prime denotes the derivative with respect to  and  is an angular frequency to be determined.
Let the periodic solution () of (3) exist and assume that it can be expressed by such a set of base functions According to (4), the initial approximate periodic solution satisfying initial conditions should be where  0 is an unknown constant to be determined later.
Here, we will use the residual to improve the accuracy.Substituting (5) into (3), the initial residual can be obtained: If  0 () = 0, then  0 () happens to be the exact solution.However, in general, such case will not appear for nonlinear problems.
According to harmonic balance method, no secular term in (6) requires eliminating contributions proportional to cos , through which the unknown constant  0 can be determined.Then, the zero-order approximation solution is obtained of the form with initial residual In the following, we introduce a bookkeeping parameter  with values in the interval [0, 1], denote the (),  as (, ), (), and then expand (, ) and in which Obviously, when  = 0, (, 0) is the zeroth-order approximation and when  = 1, (, 1) is the required approximate solution of (3).Generally, ( 9) provides the higher-order approximation to the exact solution.For example, the firstorder analytical approximation turns out to be Substituting ( 11) into (3) and equating the coefficients of the  yield By considering (4) and harmonic balance method, let Substituting ( 13) into (12), we consider the following equation: in this way, the initial residual  0 () is introduced into (14) to improve the accuracy.
According to harmonic balance method, the right-hand side of ( 14) should not contain the terms cos , cos 3.Letting their coefficients be zeros yields two linear equations with two unknowns  1 and  3,1 , through which the two unknowns can be solved easily.We now get the first-order approximation with the residual where  0 () and  1 () are given by ( 7) and ( 13), respectively.

Solution Procedure
By introducing a new time scale  = , then (1) becomes where prime denotes the derivative with respect  and  is an angular frequency to be determined.By following (7), the initial approximation with the initial conditions can be written into Here, it is possible to do the following Fourier series expansion: in which Substituting ( 23) and ( 24) into ( 22) yields in which Eliminating contributions proportional to cos  in (26), that is, solving equation  01 = 0, gives which is the same as the one by Xu [24].Till now we get the zeroth-order analytical approximation (23) and the initial residual To obtain the first-order analytical solution, we substitute (11) and ( 13) into (12), expand the function (cos  − cos 3)(cos ) −2/3 into a Fourier series Mathematical Problems in Engineering According to (16), we have Setting the coefficients of cos , cos 3 to zero in the righthand side of (34) gives Thus, we obtain the first-order approximation with the residual where  0 ,  1 , and  3,1 are determined by ( 28), (36), and (37), respectively.
With the procedure going on, similarly, we can get the high-order approximation.For example, the second-order approximation can be obtained as follows: in which The variables , , , , and  are presented in Appendix.

Results and Discussion
In order to illustrate and verify the efficiency and correctness of the presented approach for this strongly nonlinear oscillator, we consider some special cases.
Then, from (41), the second-order approximation of the frequency can be obtained: The exact frequency [16] is  = 1.070451 −1/3 .Therefore, it can be easily proved that the maximal relative error is less than 0.032%.
Hence, from (40), the second-order approximate solution can be expressed as follows: which agrees very well with the exact solution [6] as shown in Figure 1.
To further illustrate and verify the accuracy of the presented approach in this case, we present the comparison between the approximate and exact frequencies for the second-order approximation by using different methods in Table 1.It is clear that, for the second-order approximation, the result obtained in this paper is better than those obtained previously by other authors.
In this case, the first-and second-order approximate periods obtained by (38) and (41) for different values of  2 are shown in Table 2.The relative errors (RE) are defined as RE = |( −  ex )/ ex | × 100.It can be observed that the approximate periods have a good adjustment with the exact ones.
To verity results, Figure 2 shows the comparisons of the second-order analytical solutions obtained by (40) with the exact ones for  = 0.1, 1, 10, and 100 when  = 1.It can be seen from this figure that our analytical results are very close to the exact ones for the wide range of initial amplitude in this case.
Case 3. If  = 1,  = 0,  = 1, we can obtain the following nonlinear oscillator: From Table 3 and Figure 3, we can see that the accuracy of the results obtained in this paper is in excellent agreement with exact ones for the wide range of initial amplitude .

Conclusions
An iterative homotopy harmonic balance approach has been presented and applied to deduce the accurate approximations to the angular frequency and periodic solution of a strongly nonlinear oscillator.The high-order analytical approximations of the frequency and solution of this oscillator are obtained.Excellent agreement between approximate results by this approach and the exact ones has been demonstrated and discussed; the discrepancy between the second-order approximate results and exact ones is very very low.And, we can see that the approach considered here is very simple in its principle and has great potential to be applied to other nonlinear oscillators.

Table 1 :
Comparison of the exact and approximate frequencies (2nd-order) obtained by using different methods.

Table 2 :
Comparison of approximate periods with the exact period for Case 2.

Table 3 :
Comparison of approximate periods with the exact period for Case 3.