Application of Homotopy Perturbation Method with an Auxiliary Term for Nonlinear Dropping Equations Arisen in Polymer Packaging System

Dropping is an unavoidable situation for a packaged product while delivered, which is investigated by many researchers [1–3]. In most cases, the constitutive models of cushioning package materials are strong nonlinear. Various kinds of nonlinear oscillation problems exist in the engineering field, which are usually difficult to be solved analytically. However, the analytical solution is more significant for the further intensive study. Among the methods for analytical solution, the perturbation method [4] is one of the most well-known approaches, and it is based on the existence of small parameters which are not commonly existed in many nonlinear problems. Besides, in order to avoid some restrictions of perturbation method, some other methods are developed, including the variational iteration method (VIM) [5], the homotopy analysismethod (HAM) [6], He’smax-min method, and the homotopy perturbation method (HPM) [7– 11]. HPM is an analytical method providing an alternative approach to introducing an expanding parameter and applied to many areas of science and engineering [12–17]. Nawaz [18] studied nonlinear boundary value problems for fourth-order fractional integrodifferential equations using both VIM and HPM, and the comparison results showed that both methods were very effective and convenient to solve this problem. Noor [19] modified the HPM with an auxiliary term which makes the HPM more flexible. In recent studies, He [20] summarized the modification of the HPM by introducing an auxiliary term in the homotopy equation, and Duffing equation was used as an example to illustrate the solution procedure. Polymer foams, especially expanded polystyrene (EPS), are widely used for cushion or protective inner packaging, and the governing equations [21] can be expressed as


Introduction
Dropping is an unavoidable situation for a packaged product while delivered, which is investigated by many researchers [1][2][3].In most cases, the constitutive models of cushioning package materials are strong nonlinear.
Various kinds of nonlinear oscillation problems exist in the engineering field, which are usually difficult to be solved analytically.However, the analytical solution is more significant for the further intensive study.Among the methods for analytical solution, the perturbation method [4] is one of the most well-known approaches, and it is based on the existence of small parameters which are not commonly existed in many nonlinear problems.Besides, in order to avoid some restrictions of perturbation method, some other methods are developed, including the variational iteration method (VIM) [5], the homotopy analysis method (HAM) [6], He's max-min method, and the homotopy perturbation method (HPM) [7][8][9][10][11].HPM is an analytical method providing an alternative approach to introducing an expanding parameter and applied to many areas of science and engineering [12][13][14][15][16][17].Nawaz [18] studied nonlinear boundary value problems for fourth-order fractional integrodifferential equations using both VIM and HPM, and the comparison results showed that both methods were very effective and convenient to solve this problem.
Noor [19] modified the HPM with an auxiliary term which makes the HPM more flexible.In recent studies, He [20] summarized the modification of the HPM by introducing an auxiliary term in the homotopy equation, and Duffing equation was used as an example to illustrate the solution procedure.
Polymer foams, especially expanded polystyrene (EPS), are widely used for cushion or protective inner packaging, and the governing equations [21] can be expressed as where  denotes the displacement of the product while dropping, m; the coefficient  denotes the mass of the packaged product, kg;  1 and  2 are the displacement impendence, m −1 ;  3 ,  4 , and  5 are the elasticity, N, and  i denote, respectively, the characteristic constants of polymer foams which could be obtained by compression test;  is the acceleration of gravity, m/s −2 ; and ℎ is the dropping height, m.The polymer packaging system can be shown in Figure 1.By introducing these parameters:  0 = √/ 1  3 ,  = 1/ 1 ,  1 =  2 / 1 ,  2 =  4 / 3 , and  3 =  5 / 3 , and letting  = /,  = / 0 , (1) can be written in the following nondimensional forms:

Mathematical Problems in Engineering
This paper investigated for the first time the applicability and the validity of the modified HPM for EPS polymer cushioning packaging system.Besides, in order to show the accuracy of this method, some specific parameters were used in the constitutive equation based on real situation, and solutions of the modified HPM and Runge-Kutta method were compared.

Homotopy Perturbation Method with an Auxiliary Term
Considering a general nonlinear equation where  and  are the linear operator and nonlinear operator, respectively.
According to the classic HPM, the homotopy equation can be constructed as where L, constructed based on the model property, is a linear operator and L = 0 can depict the solution property.With the increase of the embedding parameter  from 0 to 1, (4) will transform from L = 0 to the original one.
According to He's recent study [20], the homotopy equation can be constructed with an auxiliary term as where  is the auxiliary parameter.It is obvious that when  = 0, (5) is equivalent to the classical equation ( 4).And  can be determined in the iteration procedure.The solutions of both ( 4) and ( 5) can be expanded into a series in :

Nonlinear Polymer Packaging System
By the Taylor series, (2) can be expanded by Taylor series as follows to simplify the calculation: According to [16], the homotopy equation with an auxiliary term can be constructed as According to the classical perturbation method, the iteration equations can be constructed as Solve (9) and then obtain the initial approximate solution where  and Ω are, respectively, the displacement amplitude and the frequency, and can be further determined.Substitute ( 12) into (10) and rewrite it as where In order to eliminate the secular term, it must be satisfied that  1 = 0. Thus, A special solution of the ordinary differential equation ( 13) can be easily obtained as Thus, by substituting ( 16) into ( 11), the second-order iteration equation can be expressed as where And no secular terms require  1 = 0; thus By the same way, the second-order iteration solution can be obtained by substituting (12) into (17): By solving ( 15) and ( 19) simultaneously, we can obtain the frequency Ω and the auxiliary term .And the final solution  can be uniformed by  =  0 +  1 +  2 .

Results
In order to verify the above method, the approximate solution by the new HPM was compared with the numerical solution solved by the Runge-Kutta method, as illustrated in Figure 2 with parameters:  = 1,  1 = 2,  2 = 10, and  3 = 20.The results showed a good agreement.
Different parameters may lead to the different accuracy of the HPM solution.Table 1 showed the accuracy of HPM  compared with the numerical solution in different  1 ,  2 , and  3 , and the results showed that, for different parameters, the HPM solutions were all in good agreement with the numerical solutions which can be almost equal to the exact solution.

Conclusion
It is desirable to obtain the solution of strong nonlinear equation arisen in polymer packaging system.In this paper, the homotopy perturbation method with an auxiliary term was applied, and the solution was obtained and compared with the Runge-Kutta method, showing good agreement.The results indicate that the HPM with an auxiliary term was suitable for solving the strong nonlinear vibration problems in packaging system.From the comparison results shown above, it can be concluded that, for the polymer packaging system with different parameters ( 1 ,  2 , and  3 ), the HPM can be a vigorous method to obtain its frequency.What is more, different parameters will lead to different error value.
And by roughly estimate, with the increase of  1 , the HPM solution's relative error increases; with the increase of  2 and  3 , the HPM solution's relative error decreases.Thus, the future research can focus on the contribution of different  1 ,  2 , and  3 to the HPM's accuracy and to what range of  1 ,  2 , and  3 values the HPM can be applied.

𝑥:
The displacement of the product while dropping, m : Th em a s so ft h ep a c k a g e dp r o d u c t ,k g  1 and  2 : The displacement impendence, m −1  3 ,  4 , and  5 : The elasticity, N : The acceleration of gravity, m/s −2 ℎ: The dropping height, m  0 , , and  i : Nondimensionalized parameters : Nondimensionalized displacement : Nondimensionalized time : Nondimensionalized velocity : Nondimensionalized displacement amplitude Ω: Nondimensionalized frequency  i and  i : Parameters used to simplify the expression : Th e a u x i l i a r y t e r m f o r H P M .

Figure 2 :
Figure 2: Comparison between the HPM solution and the numerical solution.

Table 1 :
Comparison between the HPM solution and the numerical solution.