On the Convergence of the Homotopy Analysis Method for Inner-Resonance of Tangent Nonlinear Cushioning Packaging System with Critical Components

Abstract

Homotopy analysis method (HAM) is applied to obtain the approximate solution of inner-resonance of tangent cushioning packaging system based on critical components. The solution is obtained in the form of infinite series with components which can be easily calculated. Using a convergence-control parameter, the HAM utilizes a simple method to adjust and control the convergence region of the infinite series solution. The obtained results show that the HAM is a very accurate technique to obtain the approximate solution.

1. Introduction

One of the most important subjects in packaging system is to investigate products damaged due to being dropped. Many researchers have investigated cushioning packaging system in this special field [1, 2]. The mechanical and electronic products are composed of large number of elements and, generally, damage at the so-called critical components. To prevent any damage, a critical component and a cushioning packaging are included in a package system [3]. The following assumptions are made basically in the last decade by the researchers [1, 4]. (1) The researchers considered that the packaging system is a spring-mass, single degree of freedom system. (2) The use of simple linear or nonlinear springs for cushioning packaging may not be appropriate. Wang et al. [2] considered a linear model for this system, while the oscillation in the package system is inborn nonlinearity (see [4, 5]). Our goal of this paper is to obtain the approximate solution of inner-resonance of tangent nonlinear cushioning packaging system with critical components introduced in [1, 4] using HAM, which is one of the semilinear approximate analytical methods.

In the last two decades, many researchers have employed the approximate analytical methods such as adomian decomposition method (ADM), variational iteration method (VIM), homotopy perturbation method (HPM), and HAM to solve differential equations. These methods give the solution of the differential equations in the form of infinite series. One of the advantages of approximate analytical methods is that these methods do not produce rounding-off errors. Contrary to the implicit finite difference method (FDM), the approximate analytical methods do not require the numerical solution of the systems of differential equations.

We apply the HAM to construct the series solution for the inner-resonance of tangent nonlinear cushioning packaging system with critical components. An advantage of HAM over perturbation methods is that it is not dependent on small or large parameters. As it is well known, perturbation methods cannot be applied to all nonlinear equations because these methods are based on the existence of small or large parameters. Besides, nonperturbation methods are independent of small parameters. According to [6], both of the two techniques (perturbation and nonperturbative techniques) cannot provide a simple procedure to adjust or control the convergence region and rate of given approximate series. According to [7, 8], HAM also allows for fine-tuning of convergence region and rate of convergence by allowing an auxiliary parameter ħ to vary. To the best of our knowledge, this is the first attempt at solving the inner-resonance of tangent nonlinear cushioning packaging system with components approximately using the HAM.

In recent years, the HAM has been successfully applied for solving various nonlinear systems of equations in many branches of mathematics and sciences, such as strongly coupled reaction-diffusion system [9], fractional Lorenz system [7], coupled Schrodinger-KdV equation [10], Burgers and coupled Burgers equation [11], system of second-order BVPs [8], and HIV infection CD4⁺T-cell [12]. For more studies of HAM and its applications, the readers are also referred to see [13–16].

Our paper is organized as follows.

In Section 2, we introduce mathematical modelling of the inner-resonance of tangent nonlinear cushioning packaging system with critical components. In Section 3, we present a description of the HAM on system of equations, as expanded by previous researchers in particular [9, 12], applied to the inner-resonance of tangent nonlinear cushioning packaging system with critical components. We prove the convergence of homotopy series solution for the inner-resonance of tangent nonlinear cushioning packaging system with critical components also in this section. In Section 4, we have applied the HAM to obtain the approximate solution of inner-resonance of tangent nonlinear cushioning packaging system with critical components. Finally, in Section 5, we give the conclusion of this study.

2. Modeling and Equations

Generally, imagine that everything we know and have a relationship to, including things such as art, clothes, possessions, homes, gardens, trees and fields, mountains, lakes, oceans, continents, our friends, and loved ones, are instances where resonance can and does occur. Now, consider that everything we do and think is an attempt to seek out and to return to the experience of resonance, a return to the feeling of Belonging and feeling that things feel right. Even though we may not identify the motivation for making and forming certain relationships, the real attraction and value of any relationship is whether it is fulfilling and makes us feel good. We are searching for the feeling of resonance. The profound nature of these kinds of experiences is dependent on the nature and quality of the relationship we can have to something of an external nature, but we also have the potential to experience resonance within our own body and our inner being, and this I would call a “state of Inner Resonance.” When a practitioner consciously perceives resonance happening in a therapeutic relationship between themselves and patient, it is also a state of Inner Resonance. In this circumstance, the practitioners create the appropriate environment, conditions, and quality of presence so that they can receive the patients intention to be understood and received. In another way, the patient’s need to experience resonance (whether consciously or unconsciously) is met. Inner Resonance, then, is a state of receptive presence or conscious empathy between two or more people.

In Figure 1, the model of packaging system with critical component is shown to be considered as a nonlinear spring with stiffness coefficient𝑘₂. It can also idealize the joining part between the mass of critical component𝑚₁and the main part of the product𝑚₂as a linear spring with stiffness coefficient𝑘₁. According to Figures 1 and 2, the motion of this system can be written as [1, 4] 𝑚₁𝑑²𝑥(𝑡)𝑑𝑡²+𝑘₁𝑥(𝑡)(𝑡)𝑚−𝑦=0,₂𝑑²𝑦(𝑡)𝑑𝑡²+2𝑘₂𝑑_𝑏𝜋𝜋tan2𝑑_𝑏𝑦(𝑡)−𝑘₁𝑥(𝑡)−𝑦(𝑡)=0, (1) with initial conditions 𝑥(0)=0,𝑥^′(0)=𝑦(0)2𝑔ℎ,=0,𝑦^′(0)=2𝑔ℎ. (2)

Figure 1 

The model of packaging system with critical component (𝑚1≪𝑚2) and𝑥1=𝑥(𝑡),𝑥2=𝑦(𝑡).

(a)

(b)

Figure 2 

Theħ-curves of𝑋′(0.1)and𝑌′(0.1)obtained by the 5-order approximation of HAM.

Table 1 summarize the meanings of parameters and variables. To simplify (1), new variables are introduced as 𝑥𝑋=𝐿𝑦,𝑌=𝐿𝑡,𝑇=𝑇₀, (3) where 𝑇₀=𝑚₂𝑘₂,𝐿=2𝑑_𝑏𝜋. (4) We define the frequency parameters of the critical component and main part of product as𝑤₁=𝑘₁/𝑚₁and𝑤₂=𝑘₂/𝑚₂, respectively. The notations𝜆₁=𝑤₁/𝑤₂and𝜆₂=𝑚₁/𝑚₂are considered as parameter ratio and mass ratio, respectively. By considering all parameters defined, (1) can be equivalently written in the following system of nonlinear equations [1]: 𝑑²𝑋𝑑𝑇²+𝑤²₀₁𝑋−𝑤²₀₁𝑑𝑌=0,₂𝑌𝑑𝑇²+𝑤²₀₂1𝑌+3𝑌³+2𝑌15⁵+1−𝑤²₀₂𝑋=0, (5) with initial conditions 𝑋(0)=0,𝑋^′(0)=𝑇₀𝐿𝑌(0)2𝑔ℎ,=0,𝑌^′(0)=𝑇₀𝐿2𝑔ℎ, (6) where𝑋=𝑋(𝑇),𝑌=𝑌(𝑇)and 𝑤₀₁=𝜆₁,𝑤₀₂=1+𝜆²₁𝜆₂. (7)

Table 1 

List of variables and parameters (modified from [1, 4]).

3. Homotopy Analysis Method (HAM)

To apply the HAM, the nonlinear system (5) is considered. We make initial gusses on𝑋(𝑇)and𝑌(𝑇)such that they satisfy the initial conditions (6) that are defined as 𝑋(0)=𝑋₀=𝑇₀𝐿𝑌(0)2𝑔ℎ𝑇,=𝑌₀=𝑇₀𝐿2𝑔ℎ𝑇. (8) The auxiliary linear operators ℒ_𝑋 and ℒ_𝑌 are selected as ℒ_𝑋=𝑑²𝑋𝑑𝑇²,ℒ_𝑌=𝑑²𝑌𝑑𝑇², (9) satisfying the following properties: ℒ_𝑋𝑐_1,𝑋𝑇+𝑐_2,𝑋ℒ=0,_𝑌𝑐_1,𝑌𝑇+𝑐_2,𝑌=0, (10) where 𝑐_1,𝑋, 𝑐_2,𝑋, 𝑐_1,𝑌, and 𝑐_2,𝑌 are integral constants. Define the homotopy maps ℋ_𝑋𝑋,𝑌=ℒ𝑇;𝑞𝑇;𝑞1−𝑞_𝑋𝑋𝑇;𝑞−𝑋₀(𝑇)−𝑞ħ𝐻_𝑋(𝑡)𝑁_𝑋𝑋,𝑌,ℋ𝑇;𝑞𝑇;𝑞_𝑌𝑋,𝑌=ℒ𝑇;𝑞𝑇;𝑞1−𝑞_𝑌𝑌𝑇;𝑞−𝑌₀(𝑇)−𝑞ħ𝐻_𝑌(𝑡)𝑁_𝑌𝑋𝑇;𝑞,𝑌𝑇;𝑞, (11) where 𝑞∈[0,1] is an embedding parameter, ħ is nonzero auxiliary parameter, 𝐻_𝑋 and 𝐻_𝑌 are auxiliary functions, and 𝑁_𝑋 and 𝑁_𝑌 are nonlinear operators that are defined as 𝑁_𝑋𝑋,𝑌=𝜕𝑇;𝑞𝑇;𝑞²𝑋𝑇;𝑞𝜕𝑇²+𝑤²₀₁𝑋𝑇;𝑞−𝑤²₀₁𝑌,𝑁𝑇;𝑞_𝑌𝑋,𝑌=𝜕𝑇;𝑞𝑇;𝑞²𝑌𝑇;𝑞𝜕𝑇²+𝑤²₀₂𝑌+𝑌𝑇;𝑞𝑇;𝑞³3+2𝑌15𝑇;𝑞⁵+1−𝑤²₀₂𝑋𝑇;𝑞. (12) Clearly, when𝑞=0, we have the homotopy maps ℋ_𝑋𝑋(𝑇;0),𝑌(𝑇;0)=ℒ_𝑋𝑋(𝑇;0)−𝑋₀(𝑇,ℋ_𝑌)𝑋(𝑇;0),𝑌(𝑇;0)=ℒ_𝑌𝑌(𝑇;0)−𝑌₀(𝑇). (13) And when𝑞=1, we have ℋ_𝑋𝑋(𝑇;1),𝑌(𝑇;1)=−ħ𝐻_𝑋(𝑇)𝑁_𝑋𝑋(),𝑌(,ℋ𝑇;1𝑇;1)_𝑌𝑋(𝑇;1),𝑌(𝑇;1)=−ħ𝐻_𝑌(𝑇)𝑁_𝑌𝑋(𝑇;1),𝑌(𝑇;1). (14) Thus, by requiring ℋ_𝑋𝑋𝑇;𝑞,𝑌𝑇;𝑞=ℋ_𝑌𝑋𝑇;𝑞,𝑌𝑇;𝑞=0, (15) we can obtain 1−𝑞ℒ_𝑋𝑋𝑇;𝑞−𝑋₀(𝑇)=𝑞ħ𝐻_𝑋(𝑇)𝑁_𝑋𝑋,𝑌,ℒ𝑇;𝑞𝑇;𝑞1−𝑞_𝑌𝑌𝑇;𝑞−𝑌₀(𝑇)=𝑞ħ𝐻_𝑌(𝑇)𝑁_𝑌𝑋𝑇;𝑞,𝑌𝑇;𝑞. (16) If𝑞=0and𝑞=1, the homotopy equations are as follows: 𝑋(𝑇;0)=𝑋₀,𝑋()(𝑇),𝑌()𝑇;1=𝑋𝑇;0=𝑌₀,𝑌(𝑇;1)=𝑌(𝑇). (17) As𝑞varies from0to1, the solution of the nonlinear system (5) will vary from the initial guesses𝑋₀(𝑇)and𝑌₀(𝑇)to the exact solutions𝑋(𝑇)and𝑌(𝑇)of the nonlinear system (5). Expanding𝑋(𝑇;𝑞)and𝑌(𝑇;𝑞)as a Taylor series with respect to𝑞yields 𝑋𝑇;𝑞=𝑋₀+^∞_𝑚=1𝑋_𝑚𝑞^𝑚,𝑌𝑇;𝑞=𝑌₀+^∞_𝑚=1𝑌_𝑚𝑞^𝑚, (18) where 𝑋_𝑚=1𝜕𝑚!^𝑚𝑋(𝑇;𝑞)𝜕𝑞^𝑚_𝑞=0,𝑌_𝑚=1𝜕𝑚!^𝑚𝑌(𝑇;𝑞)𝜕𝑞^𝑚_𝑞=0. (19) According to [17], the convergence of the series (18) strongly depends on the auxiliary parameter ħ. Note that if 𝑞=1, then 𝑋(𝑇;1)=𝑋=𝑋₀+^∞_𝑚=1𝑋_𝑚,𝑌(𝑇;1)=𝑌=𝑌₀+^∞_𝑚=1𝑌_𝑚. (20) According to definitions (18), the governing equations for the unknowns can be deduced from the zeroth-deformation equations (16). For further analysis, the vectors are defined as 𝑋_𝑛=𝑋₀,𝑋₁,…,𝑋_𝑛,𝑌_𝑛=𝑌₀,𝑌₁,…,𝑌_𝑛. (21) Differentiating (16) 𝑚-times with respect to 𝑞, dividing by 𝑚!, and setting 𝑞=0 give the linear equations ℒ_𝑋𝑋_𝑚−𝜒_𝑚𝑋_𝑚−1=ħ𝐻_𝑋(𝑇)𝑅_𝑚,𝑋𝑋_𝑚−1,𝑌_𝑚−1,ℒ_𝑌𝑌_𝑚−𝜒_𝑚𝑌_𝑚−1=ħ𝐻_𝑌(𝑇)𝑅_𝑚,𝑌𝑋_𝑚−1,𝑌_𝑚−1, (22) with initial conditions 𝑋_𝑚(0)=0,𝑋^′_𝑚(0)𝑌=0,_𝑚(0)=0,𝑌^′_𝑚(0)=0, (23) where 𝑅_𝑚,𝑋𝑋_𝑚−1,𝑌_𝑚−1=𝑋^′′_𝑚−1+𝑤²₀₁𝑋_𝑚−1−𝑤²₀₁𝑌_𝑚−1,𝑅_𝑚,𝑌𝑋_𝑚−1,𝑌_𝑚−1=𝑌^′′_𝑚−1+𝑤²₀₂𝑌_𝑚−1+13^𝑚−1_𝑖1=0𝑌_𝑖1^{𝑚−1−𝑖1}_𝑖2=0𝑌_𝑖2𝑌_{𝑚−1−𝑖1−𝑖2}+215^𝑚−1_𝑖1=0𝑌_𝑖1^{𝑚−1−𝑖1}_𝑖2=0𝑌_𝑖2^{𝑚−1−𝑖1−𝑖2}_𝑖3=0𝑌_𝑖3×^{𝑚−1−𝑖1−𝑖2−𝑖3}_𝑖4=0𝑌_𝑖4𝑌_{𝑚−1−𝑖1−𝑖2−𝑖3−𝑖4}+1−𝑤²₀₂𝑋_𝑚−1,𝜒_𝑚∶=0𝑚≤11𝑚>1. (24)(25)(26) Using𝐻_𝑋(𝑇)=𝐻_𝑌(𝑇)=1, the solution of the𝑚-order deformation equations (22) for𝑚≥1becomes 𝑋_𝑚=𝜒_𝑚𝑋_𝑚−1+ħ𝑅_𝑚,𝑋𝑋_𝑚−1(𝜏),𝑌_𝑚−1(𝜏)𝑑𝜏𝑑𝜏+𝑐_1,𝑋𝑇+𝑐_2,𝑋,𝑌_𝑚=𝜒_𝑚𝑌_𝑚−1+ħ𝑅_𝑚,𝑌𝑋_𝑚−1(𝜏),𝑌_𝑚−1(𝜏)𝑑𝜏𝑑𝜏+𝑐_1,𝑌𝑇+𝑐_2,𝑌. (27) The coefficients 𝑐_1,𝑋, 𝑐_2,𝑋, 𝑐_1,𝑌, and 𝑐_2,𝑌 are determined using initial conditions (23).

3.1. Convergence Theorem

Theorem 1. The series𝑋(𝑇)=𝑋₀+∑^∞_𝑚=1𝑋_𝑚and𝑌(𝑇)=𝑌₀+∑^∞_𝑚=1𝑌_𝑚converge where𝑋_𝑚and𝑌_𝑚are governed by (22) under definitions (24)–(26); 𝑋 and 𝑌 must be the solutions of system of (5).

Proof. If the series∑^∞_𝑚=0𝑋_𝑚and∑^∞_𝑚=0𝑌_𝑚are convergent, we can write 𝑆_𝑋=^∞_𝑚=0𝑋_𝑚,𝑆_𝑌=^∞_𝑚=0𝑌_𝑚. (28) And it holds that lim_𝑚→∞𝑋_𝑚=lim_𝑚→∞𝑌_𝑚=0. (29) From (22) and using (9), we have ħ^∞_𝑚=1𝑅_𝑚,X𝑋_𝑚−1,𝑌_𝑚−1=^∞_𝑚=1ℒ_𝑋𝑋_𝑚−𝜒_𝑚𝑋_𝑚−1=lim^𝑛_{𝑚→∞𝑚=0}ℒ_𝑋𝑋_𝑚−𝜒_𝑚𝑋_𝑚−1=ℒ_𝑋lim^𝑛_{𝑚→∞𝑚=0}𝑋_𝑚−𝜒_𝑚𝑋_𝑚−1=ℒ_𝑋lim_𝑚→∞𝑋_𝑛ħ=0,_∞𝑚=1𝑅_𝑚,𝑌𝑋_𝑚−1,𝑌_𝑚−1=^∞_𝑚=1ℒ_𝑌𝑌_𝑚−𝜒_𝑚𝑌_𝑚−1=lim^𝑛_{𝑚→∞𝑚=0}ℒ_𝑌𝑌_𝑚−𝜒_𝑚𝑌_𝑚−1=ℒ_𝑌lim^𝑛_{𝑚→∞𝑚=0}𝑌_𝑚−𝜒_𝑚𝑌_𝑚−1=ℒ_𝑌lim_𝑚→∞𝑌_𝑛=0. (30) Sinceħ≠0, then ^∞_𝑚=1𝑅_𝑚,𝑋𝑋_𝑚−1,𝑌_𝑚−1=0,_∞𝑚=1𝑅_𝑚,𝑌𝑋_𝑚−1,𝑌_𝑚−1=0. (31)(32) Substituting (24) into (31) and simplifying it, we obtain ^∞_𝑚=1𝑅_𝑚,𝑋𝑋_𝑚−1,𝑌_𝑚−1=^∞_𝑚=1𝑋^′′_𝑚−1+𝑤²₀₁𝑋_𝑚−1−𝑤²₀₁𝑌_𝑚−1=^∞_𝑚=1𝑋^′′_𝑚−1+𝑤^2∞_01𝑚=1𝑋_𝑚−1−𝑤^2∞_01𝑚=1𝑌_𝑚−1=𝑑²𝑑𝑇^2∞_𝑚=0𝑋_𝑚+𝑤^2∞_01𝑚=0𝑋_𝑚−𝑤^2∞_01𝑚=0𝑌_𝑚=𝑋^′′+𝑤²₀₁𝑋−𝑤²₀₁𝑌=0. (33) We repeat this process and substitute (25) into (32), and simplifying it, we obtain ^∞_𝑚=1𝑅_𝑚,𝑌𝑋_𝑚−1,𝑌_𝑚−1=^∞_𝑚=1𝑌^′′_𝑚−1+𝑤^2∞_02𝑚=1𝑌_𝑚−1+^∞_𝑚=11−𝑤²₀₂𝑋_𝑚−1+13^∞_𝑚=1^𝑚−1_𝑖1=0𝑌_𝑖1^{𝑚−1−𝑖1}_𝑖2=0𝑌_𝑖2𝑌_{𝑚−1−𝑖1−𝑖2}+215^∞_𝑚=1^𝑚−1_𝑖1=0𝑌_𝑖1^{𝑚−1−𝑖1}_𝑖2=0𝑌_𝑖2^{𝑚−1−𝑖1−𝑖2}_𝑖3=0𝑌_𝑖3×^{𝑚−1−𝑖1−𝑖2−𝑖3}_𝑖4=0𝑌_𝑖4𝑌_{𝑚−1−𝑖1−𝑖2−𝑖3−𝑖4}. (34)
For the first three terms of (34), we can easily conclude that ^∞_𝑚=1𝑌^′′_𝑚−1+𝑤^2∞_02𝑚=1𝑌_𝑚−1+^∞_𝑚=11−𝑤²₀₂𝑋_𝑚−1=𝑌^′′(𝑇)+𝑤²₀₂𝑌(𝑇)+1−𝑤²₀₂𝑋(𝑇). (35) For the fourth and fifth terms in (34), we have 13^∞_𝑚=1^𝑚−1_𝑖1=0𝑌_𝑖1^{𝑚−1−𝑖1}_𝑖2=0𝑌_𝑖2𝑌_{𝑚−1−i1−𝑖2}=13^∞_𝑚=1^𝑚−1_𝑖1=0𝑌_𝑖1^{𝑚−1−𝑖1}_𝑖2=0𝑌_𝑖2𝑌_{𝑚−1−𝑖1−𝑖2}=13^∞_𝑖1^∞_{=0𝑚=𝑖1+1}𝑌_𝑖1^{𝑚−1−𝑖1}_𝑖2=0𝑌_𝑖2𝑌_{𝑚−1−𝑖1−𝑖2}=13^∞_𝑖1^∞_=0𝑗=0𝑌_𝑖1^𝑗_𝑖2=0𝑌_𝑖2𝑌_𝑗−𝑖2=13^∞_𝑖1=0𝑌_𝑖1^∞𝑗_{𝑗=0𝑖2=0}𝑌_𝑖2𝑌_𝑗−𝑖2=13^∞_𝑖1=0𝑌_𝑖1^∞_𝑖2^∞_=0𝑗=𝑖2𝑌_𝑖2𝑌_𝑗−𝑖2=13^∞_𝑖1=0𝑌_𝑖1^∞_𝑖2=0𝑌_𝑖2^∞_𝑗=𝑖2𝑌_𝑗−𝑖2=13^∞_𝑚=0𝑌_𝑚³=13𝑌³(𝑇),215^∞_𝑚=1^𝑚−1_𝑖1=0𝑌_𝑖1^{𝑚−1−𝑖1}_𝑖2=0𝑌_𝑖2^{𝑚−1−𝑖1−𝑖2}_𝑖3=0𝑌_𝑖3×^{𝑚−1−𝑖1−𝑖2−𝑖3}_𝑖4=0𝑌_𝑖4𝑌_{𝑚−1−𝑖1−𝑖2−𝑖3−𝑖4}=215^∞_𝑚=1^𝑚−1_𝑖1=0𝑌_𝑖1^{𝑚−1−𝑖1}_𝑖2=0𝑌_𝑖2×^{𝑚−1−𝑖1−𝑖2}_𝑖3=0𝑌_𝑖3^{𝑚−1−𝑖1−𝑖2−𝑖3}_𝑖4=0𝑌_𝑖4𝑌_{𝑚−1−𝑖1−𝑖2−𝑖3−𝑖4}=215^∞_𝑖1=0𝑌_𝑖1^∞_𝑚=𝑖1+1^{𝑚−1−𝑖1}_𝑖2=0𝑌_𝑖2×^{𝑚−1−𝑖1−𝑖2}_𝑖3=0𝑌_𝑖3^{𝑚−1−𝑖1−𝑖2−𝑖3}_𝑖4=0𝑌_𝑖4𝑌_{𝑚−1−𝑖1−𝑖2−𝑖3−𝑖4}=215^∞_𝑖1=0𝑌_𝑖1^∞𝑗_{𝑗=0𝑖2=0}𝑌_𝑖2^{𝑚−1−𝑖1−𝑖2}_𝑖3=0𝑌_𝑖3×^{𝑚−1−𝑖1−𝑖2−𝑖3}_𝑖4=0𝑌_𝑖4𝑌_{𝑚−1−𝑖1−𝑖2−𝑖3−𝑖4}=215^∞_𝑖1=0𝑌_𝑖1^∞_𝑖2=0𝑌_𝑖2^∞_{𝑚=1+𝑖1+𝑖2}^{𝑚−1−𝑖1−𝑖2}_𝑖3=0𝑌_𝑖3×^{𝑚−1−𝑖1−𝑖2−𝑖3}_𝑖4=0𝑌_𝑖4𝑌_{𝑚−1−𝑖1−𝑖2−𝑖3−𝑖4}=215^∞_𝑖1=0𝑌_𝑖1^∞_𝑖2=0𝑌_𝑖2^∞𝑙_{𝑙=0𝑖3=0}𝑌_𝑖3×^{𝑚−1−𝑖1−𝑖2−𝑖3}_𝑖4=0𝑌_𝑖4𝑌_{𝑚−1−𝑖1−𝑖2−𝑖3−𝑖4}=215^∞_𝑖1=0𝑌_𝑖1^∞_𝑖2=0𝑌_𝑖2^∞_𝑖3=0𝑌_𝑖3×^∞_{𝑚=1+𝑖1+𝑖2+𝑖3}^{𝑚−1−𝑖1−𝑖2−𝑖3}_𝑖4=0𝑌_𝑖4𝑌_{𝑚−1−𝑖1−𝑖2−𝑖3−𝑖4}=215^∞_𝑖1=0𝑌_𝑖1^∞_𝑖2=0𝑌_𝑖2^∞_𝑖3=0𝑌_𝑖3^∞𝑘_{𝑘=0𝑖4=0}𝑌_𝑖4𝑌_𝑘−𝑖4=215^∞_𝑖1=0𝑌_𝑖1^∞_𝑖2=0𝑌_𝑖2^∞_𝑖3=0𝑌_𝑖3^∞_𝑖4^𝑘_=0𝑘=𝑖4𝑌_𝑖4𝑌_𝑘−𝑖4=215^∞_𝑖1=0𝑌_𝑖1^∞_𝑖2=0𝑌_𝑖2^∞_𝑖3=0𝑌_𝑖3^∞_𝑖4=0𝑌_𝑖4^𝑘_𝑘=𝑖4𝑌_𝑘−𝑖4=2𝑌15⁵(𝑇). (36) Thus, ^∞_𝑚=1𝑅_𝑚,𝑌𝑋_𝑚−1,𝑌_𝑚−1=𝑌^′′+𝑤²₀₂1𝑌+3𝑌³+2𝑌15⁵+1−𝑤²₀₂𝑋=0. (37) From (2) and (23), it holds that 𝑋₀=^∞_𝑚=0𝑋_𝑚(0)=𝑋₀(0)+^∞_𝑚=1𝑋_𝑚(0)𝑋=0,_′0=^∞_𝑚=0𝑋^′_𝑚(0)=𝑋^′₀(0)+^∞_𝑚=1𝑋^′_𝑚(0)=𝑇₀𝐿𝑌2𝑔ℎ,₀=^∞_𝑚=0𝑌_𝑚(0)=𝑌₀(0)+^∞_𝑚=1𝑌_𝑚(0)𝑌=0,_′0=^∞_𝑚=0𝑌^′_𝑚(0)=𝑌^′₀(0)+^∞_𝑚=1𝑌^′_𝑚(0)=𝑇₀𝐿2𝑔ℎ. (38) Thus,𝑋and𝑌satisfy the system (5) and it must be the exact solution for (5) with the initial conditions (6).

4. Example

In this section, the HAM is applied to obtain the approximate solutions of the system (5) with the initial conditions (6). We have also used symbolic software Mathematica to solve the system of linear equations (22) with the initial conditions (23). Few components of the series solutions of (20) are given as follows: 𝑋₀=𝑇₀𝑔ℎ𝜋𝑇√2𝑑_𝑏,𝑌₀=𝑇₀𝑔ℎ𝜋𝑇√2𝑑_𝑏,𝑋₁𝑌=0,₁=ħ𝜋𝑇³𝑔ℎ𝑇₀6√2𝑑_𝑏1+𝑔𝜋²𝑇²ℎ𝑚₂20𝑑²_𝑏𝑘₂+𝑔²𝜋⁴𝑇⁴ℎ²𝑚²₂210𝑑⁴_𝑏𝑘²₂,𝑋₂=ħ²𝜋𝑇⁵𝑔ℎ𝑇₀𝑘₁𝑚₂120√2𝑑_𝑏𝑘₂𝑚₁1+𝑔𝜋²𝑇²ℎ𝑚₂42𝑑²_𝑏𝑘₂+𝑔²𝜋⁴𝑇⁴ℎ²𝑚²₂756𝑑⁴_𝑏𝑘²₂,𝑌₂=ħ𝜋𝑇³𝑔ℎ𝑇₀6√2𝑑_𝑏1+ħ+ħ𝑇²2+ħ𝑇²𝑘₁20𝑘₂+𝑔ħ𝜋²𝑇⁴ℎ𝑚₂𝑘₁840𝑑²_𝑏𝑘²₂+𝑔𝜋²𝑇²ℎ𝑚₂20𝑑²_𝑏𝑘₂+11𝑔ħ𝜋²𝑇⁴ℎ𝑚₂840𝑑²_𝑏𝑘₂,⋮+⋯ (39) It is clear that the HAM series solutions (20) on depend on the convergence-control parameterħwhich provides a simple way to adjust and control the convergence of the series solutions. In fact, it is very important to ensure that the series (20) are convergent. To this end, we have plottedħ-curve of 𝑋^′(0.1) and 𝑌^′(0.1) by fifth- and sixth-order approximation of the HAM in Figures 2 and 3, respectively, for values 𝑚₁=0.01, 𝑚₂=1, 𝑘₁=0.1, 𝑘₂=0.02, 𝑔=0.8, ℎ=0.01, and𝑑_𝑏=0.9. According to these ħ-curves, it is easy to discover the valid region of convergence-control parameterħwhich corresponds to the line segment nearly parallel to the horizontal axis. For clearer presentation, these valid regions have been listed in Table 2. Furthermore, these valid regions ensure us the convergence of the obtained series. Liao [18] has pointed out that when ħ=−1, the solution obtained by the HAM is the same as the series solution obtained using HPM.

Table 2 

The admissible value of ħ derived from Figures 2 and 3.

(a)

(b)

Figure 3 

Theħ-curves of𝑋′(0.1)and𝑌′(0.1)obtained by the 6-order approximation of HAM.

Now, an error analysis is introduced to obtain the optimal value of convergence-control parameter ħ. Toward this end, we define 𝜑_𝑋(𝑇;ħ) and 𝜑_𝑌(𝑇;ħ) to be 𝑚-order approximation HAM solution as follows: 𝜑_𝑋(𝑇;ħ)=^𝑚−1_𝑗=0𝑋_𝑗,𝜑_𝑌(𝑇;ħ)=^𝑚−1_𝑗=0𝑌_𝑗. (40) We substitute (40) into nonlinear system (5) and obtain the residual error functions ER_𝑋(𝑋,𝑌,ħ₁) and ER_𝑌(𝑋,𝑌,ħ₂) as follows: ER_𝑋𝑋,𝑌;ħ₁=𝑑²𝜑_𝑋𝑇;ħ₁𝑑𝑇²+𝑤²₀₁𝜑_𝑋𝑇;ħ₁−𝑤²₀₁𝜑_𝑌𝑇;ħ₁,ER_𝑌𝑋,𝑌;ħ₂=𝑑²𝜑_𝑌𝑇;ħ₂𝑑𝑇²+𝑤²₀₂𝜑_𝑌𝑇;ħ₂+13𝜑_𝑌𝑇;ħ₂³+2𝜑15^𝑌𝑇;ħ₂⁵+1−𝑤²₀₂𝜑_𝑋𝑇;ħ₂. (41)(42) Following [19], we define the square residual error for the 𝑚-order approximation to be 𝑅𝑋ħ₁=¹₀ER_𝑋𝑋,𝑌,;ħ₁²ħ𝑑𝑇,𝑅𝑌₂=¹₀ER_𝑌𝑋,𝑌,;ħ₂²𝑑𝑇. (43)

We can obtain the values of ħ₁ and ħ₂ for which the 𝑅𝑋(ħ₁) and 𝑅𝑌(ħ₂) are minimum. The optimal values of convergence-control parameters ħ₁ and ħ₂ are determined by solving the system of equations as 𝑑𝑅𝑋ħ^∗₁𝑑ħ₁ħ=0,𝑑𝑅𝑌^∗₂𝑑ħ₂=0. (44)

In [20], several methods have been introduced to find the optimal value of ħ. In Table 3, the minimum values of 𝑅𝑋(ħ₁) and 𝑅𝑌(ħ₂) have been given with optimal values of ħ^∗₁ and ħ^∗₂ for 4-, 5-, and 6-order approximations.

Table 3 

The minimum values of 𝑅𝑋(ħ^∗) and 𝑅𝑌(ħ^∗) for various orders of approximations.

In Table 4, the absolute errors ER_𝑋 and ER_𝑌 have been calculated for various 𝑇∈(0,1) when 5- and 6-order approximation HAM solutions are considered. From the table, it can be seen that the HAM provides us with the accurate approximate solution for the inner-resonance of tangent cushioning packaging system based on critical components (5).

Table 4 

The residual errors ER_𝑋 and ER_𝑌 for various 𝑇∈(0,1).

The residual errors ER_𝑋 and ER_𝑌 have been plotted in Figures 4 and 5 for 𝑇∈(0,1) and various convergence-control parametersħ. By considering these Figures, it is to be noted that the solution obtained using HAM gives an analytical solution with high order of accuracy with few iterations.

Figure 4 

The errors of residual equation (41) using the sixth-order approximate solution for variousħand𝑇∈(0,1).

Figure 5 

The errors of residual equation (42) using the sixth-order approximate solution for variousħand𝑇∈(0,1).

5. Conclusion

The homotopy analysis method (HAM) is applied to obtain approximate solution of inner-resonance of tangent cushioning packaging system based on critical components. It is shown that the HAM solution contains the convergence-control parameter ħ, which provides a simple way to adjust and control the convergence region of the resulting infinite series. The convergence of HAM is also proved for inner-resonance of tangent cushioning packaging system based on critical components. The obtained results show that HAM is an accurate and effective technique for obtaining the approximate solution of inner-resonance of tangent cushioning packaging system based on critical components.