Research Article | Open Access
On the Convergence of the Homotopy Analysis Method for Inner-Resonance of Tangent Nonlinear Cushioning Packaging System with Critical Components
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.
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 . 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.  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 , 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 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 , fractional Lorenz system , coupled Schrodinger-KdV equation , Burgers and coupled Burgers equation , system of second-order BVPs , and HIV infection CDT-cell . 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 coefficient. It can also idealize the joining part between the mass of critical componentand the main part of the productas a linear spring with stiffness coefficient. According to Figures 1 and 2, the motion of this system can be written as [1, 4] with initial conditions
Table 1 summarize the meanings of parameters and variables. To simplify (1), new variables are introduced as where We define the frequency parameters of the critical component and main part of product asand, respectively. The notationsandare 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 : with initial conditions where,and
3. Homotopy Analysis Method (HAM)
To apply the HAM, the nonlinear system (5) is considered. We make initial gusses onandsuch that they satisfy the initial conditions (6) that are defined as The auxiliary linear operators and are selected as satisfying the following properties: where , , , and are integral constants. Define the homotopy maps where is an embedding parameter, is nonzero auxiliary parameter, and are auxiliary functions, and and are nonlinear operators that are defined as Clearly, when, we have the homotopy maps And when, we have Thus, by requiring we can obtain Ifand, the homotopy equations are as follows: Asvaries fromto, the solution of the nonlinear system (5) will vary from the initial guessesandto the exact solutionsandof the nonlinear system (5). Expandingandas a Taylor series with respect toyields where According to , the convergence of the series (18) strongly depends on the auxiliary parameter . Note that if , then 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 Differentiating (16) -times with respect to , dividing by , and setting give the linear equations with initial conditions where Using, the solution of the-order deformation equations (22) forbecomes The coefficients , , , and are determined using initial conditions (23).
3.1. Convergence Theorem
Proof. If the seriesandare convergent, we can write
And it holds that
From (22) and using (9), we have
Substituting (24) into (31) and simplifying it, we obtain
We repeat this process and substitute (25) into (32), and simplifying it, we obtain
For the first three terms of (34), we can easily conclude that For the fourth and fifth terms in (34), we have Thus, From (2) and (23), it holds that Thus,andsatisfy the system (5) and it must be the exact solution for (5) with the initial conditions (6).
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: It is clear that the HAM series solutions (20) on depend on the convergence-control parameterwhich 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 plotted-curve of and by fifth- and sixth-order approximation of the HAM in Figures 2 and 3, respectively, for values , , , , , , and. According to these -curves, it is easy to discover the valid region of convergence-control parameterwhich 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  has pointed out that when , the solution obtained by the HAM is the same as the series solution obtained using HPM.
Now, an error analysis is introduced to obtain the optimal value of convergence-control parameter . Toward this end, we define and to be -order approximation HAM solution as follows: We substitute (40) into nonlinear system (5) and obtain the residual error functions and as follows: Following , we define the square residual error for the -order approximation to be
We can obtain the values of and for which the and are minimum. The optimal values of convergence-control parameters and are determined by solving the system of equations as
In Table 4, the absolute errors and have been calculated for various 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).
The residual errors and have been plotted in Figures 4 and 5 for and various convergence-control parameters. 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.
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 , 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.
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