Homotopy Analysis Method for Nonlinear Periodic Oscillating Equations with Absolute Value Term

Based on the homotopy analysis method (HAM), an analytic approach is proposed for highly nonlinear periodic oscillating equations with absolute value terms.The nonsmoothness of absolute value terms is handled bymeans of Fourier expansion, and the convergence is accelerated by using the iterationmethod. Two typical examples which can not be solved by themethod of averaging of perturbation technique are employed to illustrate the validity and flexibility of this approach. Rather, accurate approximations are obtained using the HAM-based approach. The proposed approach has general meanings and thus can be used to solve many highly nonlinear periodic oscillating systems with this type of nonsmoothness of absolute value term.


Introduction
When a periodic oscillating system is governed by a nonlinear differential equation with terms of absolute value, it is hard to gain its analytic approximation of period  by perturbation methods [1,2].This study is important because such kinds of systems are always encountered in the engineering problems.In 1982, the method of averaging is used to study the monitoring system named Lewis regulator by Hagedorn [3].In 1995, a new asymptotic method is applied to study the large amplitude rolling using only roll extinction data by Chan et al. [4].In 1999, the main theoretical and experimental aspects connected with the jumps of amplitude of rolling motion in the presence of a bifurcation have been outlined by Francescutto and Contento [5].All of the above oscillating systems contain the absolute value terms.In this paper, let us consider a type of periodic oscillating systems with absolute value term [6,7] F [ () , U () , Ü () , | ()|] = 0, (1) subject to initial conditions with  being the amplitude of (), F being a continuous operator, and dot being the differentiation with respect to .For example, consider the periodic oscillating system governed by Ü () +  2 0  () +  | ()|  () = 0,  (0) = , where  0 is the natural frequency and  is a physical parameter.The absolute term |()| brings great difficulty, because the nonlinear term becomes nonsmooth.The method of averaging [8,9] in perturbation theory is frequently used to obtain analytical approximations of (3).With the method of averaging, the 1st-order approximate frequency of (3) reads [10] as follows: Unfortunately, the above perturbation result is valid only for small amplitude of oscillation in the case of  0 = 1, as shown in  3) given by the 1storder HAM approximation with the exact result and the 1st-order perturbation approximation in the case of  0 = 1.Dash-dot: the 1st-order perturbation approximation (4).Filled circle: exact result.Solid line: the 1st-order HAM approximation (5).
problems.To overcome the restrictions of perturbation methods, Liao [11,12] developed an analytic approximation technique for highly nonlinear problems, namely, the homotopy analysis method (HAM) [13,14].The HAM, based on a basic concept in topology, transfers nonlinear problems into an infinite number of linear subproblems which are independent of any small/large physical parameters.So, this technique works efficiently for many nonlinear problems in science and engineering.In particular, the HAM provides a convenient way to guarantee the convergence of solution series.In addition, the HAM provides us with great freedom to choose the equation type of the linear sub problems so that the high-order approximations can be obtained easily in general.Due to these advantages, the HAM has been successfully applied to solve lots of different types of nonlinear problems [15][16][17][18][19][20][21][22].These successful applications indicate the validity and potential of the HAM.In many cases, the HAM can give accurate approximations even for problems with high nonlinearity.For example, using the HAM, Liao [11] gave the 1st-order approximation of frequency of (3) in the case of  0 = 1: which is accurate in the whole interval 0 ≤  < +∞, as shown in Figure 1.It should be noticed that the 1st-order perturbation result (4) is valid only for small , as shown in Figure 1, and the perturbation result (4) given by the method of averaging is invalid for high nonlinearity.
Besides, when  0 = 0, the perturbation result (4) has no meaning at all.Thus, when  0 = 0, the nonlinearity of (3) is so high that the method of the averaging of perturbation theory does not work.
Recently, based on the HAM combined with Wu's elimination method, a Maple package NOPH is developed by Liu et al. [22] to solve analytic approximations of nonlinear periodic oscillating systems.It is free and available online (http://numericaltank.sjtu.edu.cn/NOPH.htm)with many examples of applications.But this method only works for differentiable nonlinear oscillating systems and can not be used to solve nonlinear periodic oscillating systems like (1).
The current work is an extension of the HAM to solve the nonlinear oscillating systems (1) by means of Fourier expansion and iteration technique.
In this paper, the HAM based on Fourier expansion and iteration will be described in Section 2.Then, the newly developed approach is verified by solving two examples in Section 3 and the validity and the convergence of this approach are also discussed.The conclusions are given in Section 4.

Basic Ideas of the HAM Approach
there is Then, (1) can be rewritten as Assume that there is a real-valued function set {  ()} ∞ =1 with the property of uniform convergence; that is, lim  → ∞   () = (), and we can construct a family of equations: with It is obvious that ( 9) and ( 10) approach ( 1) and ( 2) as  → ∞.
As soon as an appropriate initial guess of the solution is chosen, the expression of sgn( −1 ()) is known when solving (9) using the iteration approach.In this way, the problem of (1) and ( 2) is transformed to ( 9) and (10).In the subsection, the HAM combined with the Fourier expansion method will be proposed to solve ( 9) and (10).

HAM with Fourier Expansion.
Assuming that { 0 (),  1 (), . . .,  −1 ()} has been obtained, we consider the th nonlinear subsystem of ( 9) and (10).Due to considering the periodic solution of the oscillations, the transformation is introduced, where   is the frequency of the   ().Under the transformation ( 11), ( 9) and (10) become where the dot denotes the differentiation with respect to .Note that ( 12) represents a periodic motion with the period of 2.For simplicity, only the cosine-type periodic oscillating systems are considered in this paper.Therefore,   () can be expressed by such a set of basis functions That is, where   are coefficients.
The HAM provides us with great freedom in choosing initial guess of solutions and auxiliary linear operators [11].Considering the initial conditions ( 13) and the rule of solution expression (15), the initial guess solution can be chosen as Based on the type of the governing equation and the solution expression (15), the auxiliary linear operator is chosen as Note that the operator L has the property for any constant coefficients  1 and  2 .
According to (12), a nonlinear operator is defined as The so-called zeroth-order deformation equation is constructed as subject to the initial conditions where  0 is the so-called nonzero convergence-control parameter and  is an embedding parameter.
When  = 0, ( 20) and ( 21) have the solution When  = 1, ( 20) and ( 21) are the same as ( 12) and ( 13), respectively, provided Thus, as the embedding parameter  increases from 0 to 1, the solution Φ  (; ) of the zeroth-order deformation equation deforms from the initial guess  ,0 () to the exact solution   (), and so does Ω  () from the initial guess  ,0 to the exact frequency   , respectively.From Taylor's theorem, Φ  (; ) and Ω  () are expanded in the power series of  as follows: where Assume that the value of  0 is properly chosen to the point that (24) are convergent at  = 1.Then, due to (23), one has For the simplicity, two vectors are defined as By differentiating the zeroth-order deformation equations ( 20) and ( 21)  times with respect to , then dividing them by !, and finally setting  = 0, the so-called th-order deformation equation is obtained as subject to the initial conditions where (31) Note that  , is known when the th-order deformation equation is solved.Therefore,  , () can be obtained by solving the linear equations ( 28) and (29).
Then, the general solution of (28) reads where and the integral coefficients are determined by the initial condition (29).Note that  ,−1 is unknown so far.So, one additional algebraic equation is needed to determine  ,−1 .Equation (28) can be rewritten as According to the linear auxiliary operator (17), if  ,−1,1 ̸ = 0,  , () contains the so-called secular term  sin(), which is not periodic.Such kind of secular terms must be avoided.Thus,  ,−1,1 = 0 is enforced, which provides an additional algebraic equation for  ,−1 .At this point,  ,0 ,  ,1 , . . .,  ,−1 can be obtained in sequence.In this way, the linear deformation equations ( 28) and (29) can be successively solved.
The th-order approximations of   () and   are defined as respectively.Obviously, the th-order homotopy-approximation of   () satisfies the initial condition (13).It is usually much better than the initial approximation  ,0 (), if the convergence-control  0 is properly chosen.In order to choose a proper value of  0 , the squared residual error is defined as follows: For the sake of computational efficiency, the squared residual error E  is calculated numerically where Δ = 2/ and  is an integer. = 50 is set in this paper.Given the initial guess  ,0 () and the auxiliary linear operator L, the discrete squared residual error E  is only dependent on the convergence-control parameter  0 , whose optimal value is determined by the minimum of E  .Unlike other analytical methods, the HAM provides a convenient way to guarantee the convergence of the series solution [22,23].

Examples
In this section, the validity of the proposed iteration approach is illustrated by two examples.
Example 1. First, consider the following equation: According to (11) and ( 12), (42) becomes And the first initial guess of the solution is given as follows:  The term  , in (28) becomes By using computer algebra software Mathematica, approximate solutions of the high-order deformation equations can be obtained.
When  = 1, the 1st-order and 2nd-order approximations of frequency  can be sequentially obtained: Here, some supplemental statements are needed to elaborate the origin of the piecewise function  0 ().Figure 2 illustrates that the optimal convergence-control parameter  0 can be chosen as −117/100 by plotting residual curves in the case of  = 1,  = 1.Similarly, in the case of  = 1, the following data set )} (48) can be obtained.Further, the piecewise function  0 () can be obtained easily by using the Mathematica built-in function Fit.Then,   = |( 1, −   )/  | ( = 0, 1) is defined to measure the relative error between  1, , obtained by HAM, and the numerical frequency   , listed in Table 1, which shows that  1,1 is valid even with a large amplitude such as  = 1000.Both Table 1 and Figure 3 verify the effectiveness of the frequency obtained by HAM.The optimal convergencecontrol parameter  0 = −119/100 is chosen according to the Mathematica built-in function NMinimize[E 1 ,  0 ] in the case of  =  = 1.Without loss of generality, let  = 3, that is, a 3rd-order homotopy-iteration approach applied.Thus, the solutions  1,1 (),  1,2 (), . . .,  1,−1 () are obtained by the initial guess of the solution  1,0 () =  0 = cos().Next, the initial guess of the solution will be updated by  1 = ∑  =0  1, () and a range of new solutions  2,1 (),  2,2 (), . . .,  2,−1 () will be obtained again, which can be called th-order and one iteration.If we loop  times, this process will be called th-order and  times iteration.Besides, let  = 20, and  , is approximated by the first 20 lower-frequency terms only.The homotopyapproximation at each iteration, the corresponding discrete square residual, and the homotopy-approximations of frequency   are listed in Table 2. Table 2 shows that the square residual of the homotopy-approximation decreases to 10 −8 and the frequency  of the oscillating system (42) converges to the value 0.9146812713 as the iteration increases.In Figure 4, the homotopy-approximation of () after the 10 times iteration is compared with the numerical result.It shows that the homotopy-approximation and the numerical result agree very well.According to (11) and ( 12), (49) becomes The solving procedures are the same as those defined in Section 2, except that the term  , in (28) becomes The basic idea of choosing the suitable values of parameters  =  = 1,  = 20,  = 3, and  0 = −4/25 and computing the approximations at the first several orders by using Mathematica software is similar to Example 1.The homotopyapproximation at each iteration, the corresponding discrete  square residual, and the homotopy-approximations of frequency   are listed in Table 3.It shows that the discrete square residual of the homotopy-approximation decreases to 10 −5 and the frequency  of the oscillating system (49) converges to the value 1.15614 as the iteration times  increases to 8.This illustrates that the frequency  of the nonlinear oscillating system (49) can be well approximated.As shown in Figure 5, the equilibrium point of the nonlinear oscillating system deviates from 0, which causes the slow convergence of the homotopy-series solution.The homotopyapproximation of () by the 3rd-order and 10 times iteration is compared with the numerical result in Figure 5.It shows that the homotopy-approximation and the numerical result are in well agreement.

Conclusions
Based on the homotopy analysis method, an analytic approach is proposed for highly nonlinear periodic oscillating equations containing absolute value terms.The nonsmoothness of absolute value terms is handled by of Fourier expansion, and the convergence is accelerated by means of iteration approach.Two typical examples are employed to illustrate the validity and flexibility of this approach.Note that they can not be solved by means of the method of averaging in perturbation theory.Rather, accurate approximations are obtained using the HAM-based approach proposed in this paper.This approach has general meanings and thus can be used to solve many highly nonlinear periodic oscillating systems with this type of nonsmoothness in science and engineering.

2. 1 .
Iteration Approach.By using the sign function sgn

Table 1 :
Comparison of  1,0 and  1,1 with   under different amplitude  in the case of  = 1.

Table 2 :
The discrete square residual E  and the homotopyapproximations of frequency   in the case of  =  = 1 by the 3rd-order homotopy-iteration approach with normal initial approximation  0 () =  cos() and the optimal convergencecontrol parameter  0 = −(119/100).

Table 3 :
The discrete square residual E  and the homotopyapproximations of frequency   in the case of  =  = 1 by the 3rd-order homotopy-iteration approach with normal initial approximation  0 () =  cos() and the optimal convergencecontrol parameter  0 = −(4/25).