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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 827901, 30 pages
http://dx.doi.org/10.1155/2012/827901
Research Article

Homotopy Perturbation Method and Variational Iteration Method for Harmonic Waves Propagation in Nonlinear Magneto-Thermoelasticity with Rotation

1Math. Department, Faculty of Science, Zagazig University, Zagazig 44519, Egypt
2Math. Department, Faculty of Science, Taif University, Saudi Arabia
3Math. Department, Faculty of Science, SVU, Qena 83523, Egypt
4Math. Department, Faculty of Science, El-Minia University, Egypt

Received 17 August 2011; Accepted 3 October 2011

Academic Editor: Cristian Toma

Copyright © 2012 Khaled A. Gepreel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The homotopy perturbation method and variational iteration method are applied to obtain the approximate solution of the harmonic waves propagation in a nonlinear magneto-thermoelasticity under influence of rotation. The problem is solved in one-dimensional elastic half-space model subjected initially to a prescribed harmonic displacement and the temperature of the medium. The displacement and temperature are calculated for the methods with the variations of the magnetic field and the rotation. The results obtained are displayed graphically to show the influences of the new parameters and the difference between the methods' technique. It is obvious that the homotopy perturbation method is more effective and powerful than the variational iteration method.

1. Introduction

In the past recent years, much attentions have been devoted to simulate some real-life problems which can be described by nonlinear coupled differential equations using reliable and more efficient methods. The nonlinear coupled system of partial differential equations often appear in the study of circled fuel reactor, high-temperature hydrodynamics, and thermoelasticity problems, see [14]. From the analytical point of view, lots of work have been done for such systems. With the rapid development of nanotechnology, there appears an ever-increasing interest of scientists and researchers in this field of science. Nanomaterials, because of their exceptional mechanical, physical, and chemical properties, have been the main topic of research in many scientific publications. Wave generation in nonlinear thermoelasticity problems has gained a considerable interest for its utilitarian aspects in understanding the nature of interaction between the elastic and thermal fields as well as for its applications. A lot of applications was paid on existence, uniqueness, and stability of the solution of the problem, see [57].

Much attention has been devoted to numerical methods, which do not require discretization of space-time variables or linearization of the nonlinear equations, among which the variational iteration method (VIM) suggested in [820] shows its remarkable merits over others. The method was successfully applied to a nonlinear one dimensional coupled equations in thermoelasticity [21], revealing that the method is very convenient, efficient, and accurate. The basic idea of variational iteration method is to construct a correction functional with a general Lagrange multiplier which can be identified optimally via variational theory.

The homotopy perturbation method [8, 22] has the merits of simplicity and easy execution. Unlike the traditional numerical methods, the HPM does not need discretization and linearization. Most perturbation methods assume that a small parameter exists, but most nonlinear problems have no small parameter at all. Many new methods have been proposed to eliminate the small parameter. Recently, the applications of homotopy theory among scientists appeared, and the homotopy theory becomes a powerful mathematical tool, when it is successfully coupled with perturbation theory. Sweilam and Khader [1] investigated variational iteration method for one dimensional nonlinear thermoelasticity. Applying He’s variational iteration method for solving differential-difference equation is discussed by Yildirim [23]. Noor and Mohyud-Din [24], Mohyud-Din et al. [2527] used He’s polynomials or Padé approximants to solve solving higher-order nonlinear boundary value problems, second-order singular problems, and nonlinear boundary value problems. Mohyud-Din et al. [28] applied the modified variational iteration method for free-convective boundary-layer equation using Padé approximation. Mohyud-Din and Noor [29, 30] used Homotopy perturbation method for solving some new boundary value problems. Mohyud-Din et al. [31] investigated some relatively new techniques for nonlinear problems.

In this paper, the homotopy perturbation method and variational iteration method are used to solve the coupled harmonic waves nonlinear magneto-thermoelasticity equations under influence of rotation. The Maple and Mathematica software packages are used to obtain the approximate solutions in one-dimensional half-space. The displacement and temperature which obtained have been calculated numerically and presented graphically.

2. Basic Idea of He’s Homotopy Perturbation Method

We illustrate the following nonlinear differential equation [8, 22]:𝐴(𝑢)𝑓(𝑟)=0,𝑟Λ,(2.1) with the boundary conditions:𝐵𝑢,𝜕𝑢𝜕𝑛=0,𝑟Γ,(2.2) where 𝐴 is a general differential operator, 𝐵 is a boundary operator, 𝑓(𝑟) is an analytic function, and Γ is the boundary of the domain Λ. Generally speaking, the operator 𝐴 can be divided into two parts which are 𝐿 and 𝑁, where 𝐿 is linear operator but 𝑁 is nonlinear operator. Equation (2.1) can therefore be rewritten as follows:𝐿(𝑢)+𝑁(𝑢)𝑓(𝑟)=0.(2.3) By the homotopy technique, we construct a homotopy 𝑉(𝑟,𝑝): Λ×[0,1]𝑅 which satisfies𝐻𝐿𝑢(𝑉,𝑝)=(1𝑝)(𝑉)𝐿0[𝐴]+𝑝(𝑉)𝑓(𝑟)=0,𝑟Λ,(2.4) or𝐻𝑢(𝑉,𝑝)=𝐿(𝑉)𝐿0𝑢+𝑝𝐿0+𝑝(𝑁(𝑉)𝑓(𝑟))=0,𝑟Λ,(2.5) where 𝑝[0,1] is an embedding parameter and 𝑢0 is an initial approximation of (2.1) which satisfies the boundary conditions (2.2). Obviously, from (2.4) and (2.5) we have𝐻𝑢(𝑉,0)=𝐿(𝑉)𝐿0=0,𝐻(𝑉,1)=𝐴(𝑉)𝑓(𝑟)=0.(2.6) The changing process of 𝑝 from zero to unity is just that of 𝑉(𝑟,𝑝) from 𝑢0(𝑟) to 𝑢(𝑟). In topology, this is called deformation, and 𝐿(𝑉)𝐿(𝑢0) and 𝐴(𝑉)𝑓(𝑟) are called homotopy. According to the homotopy perturbation method, we can first use the embedding parameter “𝑝” as a small parameter and assume that the solution of (2.4) and (2.5) can be written as a power series in “𝑝” as follows:𝑉=𝑉0+𝑝𝑉1+𝑝2V2+.(2.7) On setting 𝑝=1 results in the approximate solution of (2.3), we have𝑢=lim𝑝1𝑉=𝑉0+𝑉1+𝑉2+.(2.8) The combination of the perturbation method and the homotopy method is called the homotopy perturbation method, which has eliminated the limitations of the traditional perturbation methods. On the other hand, this technique can have full advantage of the traditional perturbation techniques. The series (2.8) is convergent to most cases. However, the convergent rate depends on the nonlinear operator 𝐴(𝑉).(1)The second derivative of 𝑁(𝑉) with respect to 𝑉must be small because the parameter may be relatively large, that is, 𝑝1.(2)The norm of 𝐿1(𝜕𝑁/𝜕𝑉) must be smaller than one so that the series converges.

3. Application of Homotopy Perturbation Method on the Nonlinear Magneto-Thermoelastic with Rotation Equations

In this section, we use the homotopy perturbation method to calculate the approximate solutions of the following nonlinear magneto-thermoelastic with rotation equations:1+𝜎1𝑢𝑡𝑡+Ω𝑢𝑡𝑢𝑥𝑥1𝜎2+2𝛾𝑢𝑥+3𝛿𝑢2𝑥𝛽1𝜃𝑥𝛽2𝜃𝑢𝑥𝑥=0,𝜃𝑎𝑢𝑥12𝑏𝑢2𝑥𝑡1+𝛼𝑢𝑥𝜃𝑥𝑥=0,(3.1) where 𝛾,𝛽1,𝛽2,𝑎,𝑏,𝛼 are arbitrary constants, 𝜎1,𝜎2 are the sensitive parts of the magnetic field, and Ω is the rotation parameter, with the initial conditions𝑢(𝑥,0)=𝜃(𝑥,0)=𝐴(1cos(𝑥)),𝑢𝑡(𝑥,0)=𝜃𝑡(𝑥,0)=0,(3.2) where 𝐴 is an arbitrary constant and the boundary conditions𝑢(0,𝑡)=𝜃(0,𝑡)=0,𝑢𝑡(0,𝑡)=𝜃𝑡(0,𝑡)=0.(3.3) To investigate the traveling wave solution of (3.1), we first construct a homotopy perturbation method as follows:(1𝑝)1+𝜎1𝑉𝑡𝑡𝑉0𝑡𝑡+𝑝1+𝜎1𝑉𝑡𝑡+Ω𝑉𝑡𝑉𝑥𝑥1𝜎2+2𝛾𝑉𝑥+3𝛿𝑉2𝑥𝛽1Θ𝑥𝛽2Θ𝑉𝑥𝑥Θ=0,(1𝑝)𝑡Θ0𝑡Θ+𝑝𝑡𝑎𝑉𝑥𝑡𝑏𝑉𝑥𝑉𝑥𝑡Θ𝑥𝑥𝛼𝑉𝑥𝑥Θ𝑥𝛼𝑉𝑥Θ𝑥𝑥=0,(3.4) where the initial approximations take the following form:𝑉0(𝑥,𝑡)=𝑢0Θ(𝑥,𝑡)=𝑢(𝑥,0)=𝐴(1cos(𝑥)),0(𝑥,𝑡)=𝜃0(𝑥,𝑡)=𝜃(𝑥,0)=𝐴(1cos(𝑥)).(3.5) According to the homotopy perturbation method, we can first use the embedding parameter “𝑝” as a small parameter and assume that the solution of (3.4) can be written as a power series in “𝑝” as the following:𝑉=𝑉0(𝑥,𝑡)+𝑝𝑉1(𝑥,𝑡)+𝑝2𝑉2(𝑥,𝑡)+𝑝3𝑉3(𝑥,𝑡)+,Θ(𝑥,𝑡)=Θ0(𝑥,𝑡)+𝑝Θ1(𝑥,𝑡)+𝑝2Θ2(𝑥,𝑡)+𝑝3Θ3(𝑥,𝑡)+,(3.6) where 𝑉𝑗 and Θ𝑗, 𝑗=1,2,3, are functions to be determined.

Substituting from (3.6) into (3.4) and arranging the coefficients of “𝑝” powers, we have1+𝜎1𝑉0,𝑡𝑡+𝜎1𝑉1,𝑡𝑡3𝛿𝑉20,𝑥𝑉0,𝑥𝑥𝛽2𝑉0,𝑥𝑥Θ02𝛾𝑉0,𝑥𝑉0,𝑥𝑥+𝜎2𝑉0,𝑥𝑥𝛽1Θ0,𝑥𝑉0,𝑥𝑥𝛽2𝑉0,𝑥Θ0,𝑥+Ω𝑉0,𝑡+𝑉1,𝑡𝑡𝑝+𝜎1𝑉2,𝑡𝑡2𝛾𝑉1,𝑥𝑉0,𝑥𝑥+𝑉2,𝑡𝑡2𝛾𝑉0,𝑥𝑉1,𝑥𝑥6𝛿𝑉0,𝑥𝑉1,𝑥𝑉0,𝑥𝑥𝑉1,𝑥𝑥𝛽2𝑉1,𝑥𝑥Θ0𝛽1Θ1,𝑥𝛽2𝑉0,𝑥Θ1,𝑥𝛽2𝑉1,𝑥Θ0,𝑥𝛽2𝑉0,𝑥𝑥Θ1+Ω𝑉1,𝑡3𝛿𝑉20,𝑥𝑉1,𝑥𝑥+𝜎2𝑉1,𝑥𝑥𝑝2+𝑉3,𝑡𝑡𝛽2𝑉2,𝑥Θ0,𝑥𝛽2𝑉2,𝑥𝑥Θ02𝛾𝑉2,𝑥𝑉0,𝑥𝑥2𝛾𝑉1,𝑥𝑉1,𝑥𝑥+𝜎1𝑉3,𝑡𝑡𝛽1Θ2,𝑥6𝛿𝑉0,𝑥𝑉2,𝑥𝑉0,𝑥𝑥3𝛿𝑉21,𝑥𝑉0,𝑥𝑥+𝜎2𝑉2,𝑥𝑥+Ω𝑉2,𝑡𝑉2,𝑥𝑥3𝛿𝑉20,𝑥𝑉2,𝑥𝑥2𝛾𝑉0,𝑥𝑉2,𝑥𝑥𝛽2𝑉1,𝑥𝑥Θ1𝛽2𝑉0,𝑥𝑥Θ2𝛽2𝑉0,𝑥Θ2,𝑥6𝛿𝑉0,𝑥𝑉1,𝑥𝑉1,𝑥𝑥𝛽2𝑉1,𝑥Θ1,𝑥𝑝3Θ+=0,0,𝑡+𝑎𝑉0,𝑥𝑡𝛼𝑉0,𝑥Θ0,𝑥𝑥+Θ1,𝑡𝛼𝑉0,𝑥𝑥Θ0,𝑥2𝑏𝑉0,𝑥𝑉0,𝑥𝑡Θ0,𝑥𝑥𝑝+Θ2,𝑡Θ1,𝑥𝑥𝑎𝑉1,𝑥𝑡𝛼𝑉1,𝑥𝑥Θ0,𝑥2𝑏𝑉1,𝑥𝑉0,𝑥𝑡𝛼𝑉0,𝑥Θ1,𝑥𝑥𝛼𝑉1,𝑥Θ0,𝑥𝑥2𝑏𝑉0,𝑥𝑉1,𝑥𝑡𝛼𝑉0,𝑥𝑥Θ1,𝑥𝑝2×𝛼𝑉1,𝑥𝑥Θ1,𝑥𝛼𝑉2,𝑥𝑥Θ0,𝑥2𝑏𝑉2,𝑥𝑉0,𝑥𝑡𝛼𝑉0,𝑥𝑥Θ2,𝑥2𝑏𝑉1,𝑥𝑉1,𝑥𝑡𝑎𝑉2,𝑥𝑡+Θ3,𝑡2𝑏𝑉0,𝑥𝑉2,𝑥𝑡𝛼𝑉0,𝑥Θ2,𝑥𝑥𝛼𝑉1,𝑥Θ1,𝑥𝑥𝛼𝑉2,𝑥Θ0,𝑥𝑥Θ2,𝑥𝑥𝑝3+=0.(3.7) In order to obtain the unknowns of 𝑉𝑗 and Θ𝑗, (𝑗=1,2,3,), we construct and solve the following system considering the initial conditions (3.2):𝜎1𝑉1,𝑡𝑡3𝛿𝑉20,𝑥𝑉0,𝑥𝑥𝛽2𝑉0,𝑥𝑥Θ02𝛾𝑉0,𝑥𝑉0,𝑥𝑥+𝜎2𝑉0,𝑥𝑥𝛽1Θ0,𝑥𝑉0,𝑥𝑥𝛽2𝑉0,𝑥Θ0,𝑥+Ω𝑉0,𝑡+𝑉1,𝑡𝑡𝜎=0,1𝑉2,𝑡𝑡2𝛾𝑉1,𝑥𝑉0,𝑥𝑥+𝑉2,𝑡𝑡2𝛾𝑉0,𝑥𝑉1,𝑥𝑥6𝛿𝑉0,𝑥𝑉1,𝑥𝑉0,𝑥𝑥𝑉1,𝑥𝑥𝛽2𝑉1,𝑥𝑥Θ0𝛽1Θ1,𝑥𝛽2𝑉0,𝑥Θ1,𝑥𝛽2𝑉1,𝑥Θ0,𝑥𝛽2𝑉0,𝑥𝑥Θ1+Ω𝑉1,𝑡3𝛿𝑉20,𝑥𝑉1,𝑥𝑥+𝜎2𝑉1,𝑥𝑥𝑉=0,3,𝑡𝑡𝛽2𝑉2,𝑥Θ0,𝑥𝛽2𝑉2,𝑥𝑥Θ02𝛾𝑉2,𝑥𝑉0,𝑥𝑥2𝛾𝑉1,𝑥𝑉1,x𝑥+𝜎1𝑉3,𝑡𝑡𝛽1Θ2,𝑥6𝛿𝑉0,𝑥𝑉2,𝑥𝑉0,𝑥𝑥3𝛿𝑉21,𝑥𝑉0,𝑥𝑥+𝜎2𝑉2,𝑥𝑥+Ω𝑉2,𝑡𝑉2,𝑥𝑥3𝛿𝑉20,𝑥𝑉2,𝑥𝑥2𝛾𝑉0,𝑥𝑉2,𝑥𝑥𝛽2𝑉1,𝑥𝑥Θ1𝛽2𝑉0,𝑥𝑥Θ2𝛽2𝑉0,𝑥Θ2,𝑥6𝛿𝑉0,𝑥𝑉1,𝑥𝑉1,𝑥𝑥𝛽2𝑉1,𝑥Θ1,𝑥=0,𝑎𝑉0,𝑥𝑡𝛼𝑉0,𝑥Θ0,𝑥𝑥+Θ1,𝑡𝛼𝑉0,𝑥𝑥Θ0,𝑥2𝑏𝑉0,𝑥𝑉0,𝑥𝑡Θ0,𝑥𝑥Θ=0,2,𝑡Θ1,𝑥𝑥𝑎𝑉1,𝑥𝑡𝛼𝑉1,𝑥𝑥Θ0,𝑥2𝑏𝑉1,𝑥𝑉0,𝑥𝑡𝛼𝑉0,𝑥Θ1,𝑥𝑥𝛼𝑉1,𝑥Θ0,𝑥𝑥2𝑏𝑉0,𝑥𝑉1,𝑥𝑡𝛼𝑉0,𝑥𝑥Θ1,𝑥=0,𝛼𝑉1,𝑥𝑥Θ1,𝑥𝛼𝑉2,𝑥𝑥Θ0,𝑥2𝑏𝑉2,𝑥𝑉0,𝑥𝑡𝛼𝑉0,𝑥𝑥Θ2,𝑥2𝑏𝑉1,𝑥𝑉1,𝑥𝑡𝑎𝑉2,𝑥𝑡+Θ3,𝑡2𝑏𝑉0,𝑥𝑉2,𝑥𝑡𝛼𝑉0,𝑥Θ2,𝑥𝑥𝛼𝑉1,𝑥Θ1,𝑥𝑥𝛼𝑉2,𝑥Θ0,𝑥𝑥Θ2,𝑥𝑥=0.(3.8) Consequently, we deduce after some calculations the following results:𝑢=lim𝑝1𝑉=𝑉0+𝑉1+𝑉2+,𝜃=lim𝑝1Θ=Θ0+Θ1+Θ2+,(3.9) where 𝑉0𝑉=𝐴(1cos𝑥),1=𝑡224+4𝜎14𝛾𝐴2sin2𝑥+3𝛿𝐴3cos𝑥3𝛿𝐴3cos3𝑥+4𝛽2𝐴2cos𝑥4𝛽2𝐴2cos2𝑥4𝜎2𝐴cos𝑥+4𝛽1,𝑉𝐴sin𝑥+4𝐴cos𝑥2=𝑡4𝜎321+1283𝛽22𝐴383𝛽2𝐴283𝛾2𝐴34𝛿𝐴3+4𝐴3𝜎28𝛿+3𝜎29𝐴2𝛿2𝐴543𝐴𝜎22438𝐴+3𝐴2𝜎2𝛽24𝛽2𝐴4𝛿4cos𝑥3𝛽1𝐴+𝛿𝐴3𝛽1+43𝛾𝐴3𝛽243𝐴𝜎2𝛽1+43𝛽2𝐴2𝛽1+sin𝑥8𝛾𝐴2+8𝐴2𝜎2𝛾8𝛽2𝐴34𝛾+3𝛽2𝐴2𝛽116𝛿𝐴4𝛾+sin2𝑥12𝛾𝐴3𝛽2+3𝛿𝐴3𝛽1sin3𝑥+20𝛿𝐴4+𝛾sin4𝑥203𝐴2𝜎2𝛽2+203𝛽22𝐴3+83𝛾𝐴2𝛽1+203𝛽2𝐴2+6𝛿𝐴4𝛽2+cos2𝑥634𝛿2𝐴54𝛽22𝐴3+12𝛿𝐴3+12𝛽2𝐴4𝛿+8𝛾2𝐴312𝐴3𝜎2𝛿cos3𝑥14𝛽2𝐴4𝛿cos4𝑥454𝛿2𝐴5+𝑡cos5𝑥3𝜎321+12163Ω𝐴𝛽1163𝛽1𝐴𝜎183𝛽2𝛼𝜎1𝐴3163𝛽18𝐴3𝛽2𝐴3𝛼sin𝑥163Ω𝐴2𝛾sin2𝑥+8𝛽2𝐴3𝛼𝜎1+8𝛽2𝐴3𝛼+sin3𝑥163Ω𝐴𝜎24Ω𝐴3𝛿163Ω𝐴2𝛽2163+Ω𝐴cos(𝑥)163𝛽2𝐴2+323𝛽1𝐴2𝛼𝜎1+163Ω𝐴2𝛽2+163𝛽2𝐴2𝜎1+323𝛽1𝐴2𝛼cos2𝑥+4Ω𝐴3,Θ𝛿cos3𝑥0Θ=𝐴(1cos(𝑥)),1=𝐴2Θ𝑡𝛼sin(2𝑥)+𝐴cos(𝑥)𝑡,2=𝐴21+𝜎1𝑡3312𝛼𝐴2𝛾cos3(𝑥)+2𝛼𝐴cos2(𝑥)𝛽1+36𝛼𝐴3𝛿sin(𝑥)cos3(𝑥)+12𝛼𝐴2𝛽2sin(𝑥)cos2(𝑥)+2𝛼𝐴𝜎2sin(𝑥)cos(𝑥)24𝛼𝐴3𝛿sin(𝑥)cos(𝑥)2𝛼𝐴2𝛽2sin(𝑥)cos(𝑥)2𝛼𝐴sin(𝑥)cos(𝑥)𝛼𝐴𝛽110𝛼𝛾𝐴2cos(𝑥)4𝛼𝛽2𝐴2+𝑡sin(𝑥)222cos(𝑥)+18𝑎𝛿𝐴2sin(𝑥)cos2(𝑥)+16𝑏𝐴2𝛾sin(𝑥)cos2(𝑥)+8𝑎𝛽2𝐴sin(𝑥)cos(𝑥)+2𝑎𝜎2sin(𝑥)+2𝑎𝛽1cos(𝑥)+24𝛼2𝐴2𝜎1cos3(𝑥)36𝑏𝐴3𝛿cos4(𝑥)16𝑏𝐴2𝛽2cos3(𝑥)+48𝑏𝐴3𝛿cos2(𝑥)+4𝑏𝐴2𝛽2cos2(𝑥)+8cos2(𝑥)𝑎𝛾𝐴4cos(𝑥)2𝑏𝐴𝜎220𝛼2𝐴2cos(𝑥)+24cos(𝑥)3𝛼2𝐴2+4𝑏𝐴cos2(𝑥)4𝑏𝐴6𝑎𝛿𝐴2sin(𝑥)20𝛼𝐴sin(𝑥)cos(𝑥)2𝑎𝛽2𝐴sin(𝑥)4𝑏𝐴2𝛽212𝑏𝐴3𝛿+4𝑏𝐴𝜎28𝑏𝐴2sin(𝑥)𝛾+16𝑏𝐴2𝛽2cos(𝑥)20𝛼2𝐴2cos(𝑥)𝜎14𝑎𝛾𝐴2cos(𝑥)𝜎12𝑎sin(𝑥)+4𝑏𝐴𝛽1sin(𝑥)cos(𝑥)20𝛼𝐴sin(𝑥)cos(𝑥)𝜎1.(3.10) Now we make calculations for the results obtained by the homotopy perturbation method using the Maple software package with the following arbitrary constants:𝑎=0.5,𝐴=0.001,𝑏=0.5,𝛼=1,𝛽1=𝛽2=0.05,𝛾=1,𝛿=0.8.(3.11) The results obtained in (3.9) are displayed graphically in Figures 14.

fig1
Figure 1: Variations of the displacement 𝑢 and temperature 𝜃 for various values of the axis 𝑥 and time 𝑡 whenΩ=0.1, 𝜎1=0.2, 𝜎2=0.1.
fig2
Figure 2: Variations of the displacement 𝑢 and temperature 𝜃 for various values of the axis 𝑥 and rotation Ω when 𝑡=0.1, 𝜎1=0.2, 𝜎2=0.1.
fig3
Figure 3: Variations of the displacement 𝑢 and temperature 𝜃 for various values of the axis 𝑥 and magnetic field 𝜎1 when𝑡=0.1, Ω=0.1, 𝜎2=0.1.
fig4
Figure 4: Variations of the displacement 𝑢 and temperature 𝜃 for various values of the axis 𝑥 and magnetic field 𝜎2 when 𝑡=0.1, Ω=0.1, 𝜎1=0.1.
3.1. Special Cases
(1)If we take into our consideration the first iteration (i.e., 𝑢=𝑉0+𝑉1 and 𝜃=Θ0+Θ1). See Figures 5, 6, 7, and 8.(2)If the magnetic field and rotation are neglected, the components of the displacement 𝑢 and temperature 𝜃 take the following forms. See Figures 9 and 10.
fig5
Figure 5: Variations of the displacement 𝑢 and temperature 𝜃 for various values of the axis 𝑥 and time 𝑡 when Ω=0.1, 𝜎1=0.2, 𝜎2=0.1.
fig6
Figure 6: Variations of the displacement 𝑢 and temperature 𝜃 for various values of the axis 𝑥 and rotation Ω when 𝑡=0.1, 𝜎1=0.2, 𝜎2=0.1.
fig7
Figure 7: Variations of the displacement 𝑢 and temperature 𝜃 for various values of the axis 𝑥 and magnetic field 𝜎1 when 𝑡=0.1, Ω=0.1, 𝜎2=0.1.
fig8
Figure 8: Variations of the displacement 𝑢 and temperature 𝜃 for various values of the axis 𝑥 and magnetic field 𝜎2 when 𝑡=0.1, Ω=0.1, 𝜎1=0.1.
fig9
Figure 9: Variations of the displacement 𝑢 and temperature 𝜃 for various values of the axis 𝑥 and time 𝑡 (𝑢=𝑉0+𝑉1+𝑉2 and 𝜃=Θ0+Θ1+Θ2) when Ω=𝜎1=𝜎2=0.
fig10
Figure 10: Variations of the displacement 𝑢 and temperature Θ for various values of the axis 𝑥 and time 𝑡 (𝑢=𝑉0+𝑉1 and 𝜃=Θ0+Θ1) when Ω=𝜎1=𝜎2=0.

4. Basic Idea of Variational Iteration Method

Consider the following nonhomogeneous nonlinear system of partial differential equations:𝐿1𝑢(𝑥,𝑡)+𝑁1𝐿(𝑢(𝑥,𝑡),𝜃(𝑥,𝑡))=𝑓(𝑥,𝑡),(4.1)2𝜃(𝑥,𝑡)+𝑁2(𝑢(𝑥,𝑡),𝜃(𝑥,𝑡))=𝑔(𝑥,𝑡),(4.2) where 𝐿1, 𝐿2 are linear differential operators with respect to time, 𝑁1, 𝑁2 are nonlinear operators, and 𝑓(𝑥,𝑡), 𝑔(𝑥,𝑡) are given functions.

According to the variational iteration method, we can construct correct functionals as follows:𝑢𝑛+1(𝑥,𝑡)=𝑢𝑛(𝑥,𝑡)+𝑡0𝜆1(𝐿𝜏)1𝑢𝑛(𝑥,𝜏)+𝑁1̃𝑢𝑛(̃𝜃𝑥,𝜏),𝑛(𝜃𝑥,𝜏)𝑓(𝑥,𝜏)𝑑𝜏,(4.3)𝑛+1(𝑥,𝑡)=𝜃𝑛(𝑥,𝑡)+𝑡0𝜆2𝐿(𝜏)2𝜃𝑛(𝑥,𝜏)+𝑁2̃𝑢𝑛̃𝜃(𝑥,𝜏),𝑛(𝑥,𝜏)𝑔(𝑥,𝜏)𝑑𝜏,(4.4) where 𝜆1 and 𝜆2are general Lagrange multipliers, which can be identified optimally via variational theory [820]. The second term on the right-hand side in (4.3) and (4.4) is called the corrections, and the subscript 𝑛 denotes the 𝑛th order approximation, ̃𝑢𝑛 and ̃𝜃𝑛 are restricted variations. We can assume that the above correctional functionals are stationary (i.e., 𝛿𝑢𝑛+1=0 and 𝛿𝜃𝑛+1=0), then the Lagrange multipliers can be identified. Now we can start with the given initial approximation and by the previous iteration formulas we can obtain the approximate solutions.

5. Application of the Variational Iteration Method on the Nonlinear Magneto-Thermoelastic with Rotation Equations

According to the variational iteration method and after some manipulation of (4.3) and (4.4), the correct functionals are as follows:𝑢𝑛+1(𝑥,𝑡)=𝑢𝑛(𝑥,𝑡)+𝑡0𝜆1(𝜏)1+𝜎1𝑢𝑛,𝑡𝑡(𝑥,𝜏)+Ω̃𝑢𝑛,𝑡(𝑥,𝜏)̃𝑢𝑛,𝑥𝑥1𝜎2+2𝛾̃𝑢𝑛,𝑥(𝑥,𝜏)+3𝛿̃𝑢2𝑛,𝑥(𝑥,𝜏)𝛽1̃𝜃𝑛,𝑥(𝑥,𝜏)𝛽2̃𝜃𝑛(𝑥,𝜏)̃𝑢𝑛,𝑥(𝑥,𝜏)𝑥𝜃𝑑𝜏,𝑛+1(𝑥,𝑡)=𝜃𝑛(𝑥,𝑡)+𝑡0𝜆2𝜃(𝜏)𝑛,𝑡(𝑥,𝜏)𝑎̃𝑢𝑛,𝑥𝑡(𝑥,𝜏)𝑏̃𝑢𝑛,𝑥(𝑥,𝜏)̃𝑢𝑛,𝑥𝑡̃𝜃(𝑥,𝜏)𝑛,𝑥𝑥(𝑥,𝜏)𝛼̃𝑢𝑛,𝑥𝑥̃𝜃(𝑥,𝜏)𝑛,𝑥(𝑥,𝜏)𝛼̃𝑢𝑛,𝑥̃𝜃(𝑥,𝜏)𝑛,𝑥𝑥(𝑥,𝜏)𝑑𝜏,(5.1) where ̃𝑢𝑛 and ̃𝜃𝑛 are considered as a restricted variation, that is, 𝛿̃𝑢𝑛+1=0 and 𝛿̃𝜃𝑛+1=0. Consequently, the general Lagrange multipliers 𝜆1 and 𝜆2 take the following form:𝜆1(𝜏)=𝜏𝑡1+𝜎1,𝜆2(𝜏)=1.(5.2) By the substitution of the identified Lagrange multipliers (5.2) into (5.1), we have the following iteration relations:𝑢𝑛+1(𝑥,𝑡)=𝑢𝑛(𝑥,𝑡)+𝑡0𝜏𝑡1+𝜎11+𝜎1𝑢𝑛,𝑡𝑡(𝑥,𝜏)+Ω𝑢𝑛,𝑡(𝑥,𝜏)𝑢𝑛,𝑥𝑥1𝜎2+2𝛾𝑢𝑛,𝑥(𝑥,𝜏)+3𝛿𝑢2𝑛,𝑥(𝑥,𝜏)𝛽1𝜃𝑛,𝑥(𝑥,𝜏)𝛽2𝜃𝑛(𝑥,𝜏)𝑢𝑛,𝑥(𝑥,𝜏)𝑥𝜃𝑑𝜏,𝑛+1(𝑥,𝑡)=𝜃𝑛(𝑥,𝑡)𝑡0𝜃𝑛(𝑥,𝜏)𝑎𝑢𝑛,𝑥𝑡(𝑥,𝜏)𝑏𝑢𝑛,𝑥(𝑥,𝜏)𝑢𝑛,𝑥𝑡(𝑥,𝜏)𝜃𝑛,𝑥𝑥(𝑥,𝜏)𝛼𝑢𝑛,𝑥𝑥(𝑥,𝜏)𝜃𝑛,𝑥(𝑥,𝜏)𝛼𝑢𝑛,𝑥(𝑥,𝜏)𝜃𝑛,𝑥𝑥(𝑥,𝜏)𝑑𝜏,𝑛0.(5.3) With help of Maple or Mathematica, we get the following results:𝑢0=𝜃0𝑢=𝐴(1cos𝑥),1𝑡=221+𝜎1𝛽1sin𝑥2𝐴𝛾cos𝑥sin(𝑥)+2𝛽2𝐴cos2𝑥cos𝑥3𝛿𝐴2cos𝑥𝛽2𝐴+𝜎2cos𝑥𝛽2𝐴cos𝑥+3𝛿𝐴2cos3𝑥𝜃+𝐴(1cos𝑥),1[]𝑢=𝐴cos𝑥+2𝛼𝐴cos𝑥sin𝑥𝑡+𝐴(1cos𝑥),2=𝐴cos6(𝑥)67201+𝜎1419440𝐴5𝛿2𝛽1𝛾+5760𝐴5𝛿𝛽32247860𝐴7𝛿3𝛽219440𝐴6𝛿2𝛽2219440𝐴5𝛿2𝛽217280𝛿𝛽2𝛾2𝐴5+19440𝐴5𝛿2𝛽2𝜎2+𝐴cos2(𝑥)67201+𝜎146480𝐴5𝛿𝛽22𝛾+2025𝐴4𝛿𝛽1𝛽22+270𝐴3𝛿𝛽1𝜎2𝛽2+3240𝐴4𝛿2𝛽1𝜎238880𝐴6𝛿2𝛽2𝛾8505𝐴6𝛿3𝛽1135𝐴2𝛿𝛽1𝜎226480𝐴4𝛾𝛽2𝛿135𝐴2𝛿𝛽1+6480𝐴4𝛿𝛽2𝛾𝜎23240𝐴5𝛿2𝛽1𝛽2+270𝐴2𝛿𝛽1𝜎2270𝐴3𝛿𝛽1𝛽23240𝐴4𝛿2𝛽12160𝐴4𝛿𝛽1𝛾2+45𝐴2𝛿𝛽31𝐴cos4(𝑥)67201+𝜎147200𝐴4𝛾𝛽2𝛿+118800𝐴6𝛿2𝛽2𝛾4050𝐴4𝛿2𝛽1𝜎2+4050𝐴4𝛿2𝛽1+4050𝐴5𝛿2𝛽1𝛽2+3600𝐴4𝛿𝛽1𝛾23600𝐴4𝛿𝛽1𝛽22+7200𝐴5𝛿𝛽22𝛾+30375𝐴6𝛿3𝛽17200𝐴4𝛿𝛽2𝛾𝜎2𝐴cos3(𝑥)67201+𝜎1417280𝐴5𝛿2𝛽1𝛽21440𝐴4𝛾𝛽2𝛿93960𝐴7𝛿3𝛾+16560𝐴5𝛿𝛽22𝛾+1440𝐴4𝛿𝛽2𝛾𝜎223760𝐴6𝛿2𝛽2𝛾+1440𝐴4𝛿𝛽1𝛽22720𝐴3𝛾𝛿+1440𝐴3𝛿𝛽1𝛽2+23760𝐴5𝛿2𝛾𝜎2720𝐴3𝛿𝛾𝜎225760𝐴5𝛿𝛾323760𝐴5𝛾𝛿21440𝐴3𝛿𝛽1𝜎2𝛽2+720𝐴3𝛿𝛽21𝛾+1440𝐴3𝛾𝛿𝜎2𝐴cos5(𝑥)67201+𝜎1419440𝐴5𝛿2𝛽1𝛽2+189540𝐴7𝛿3𝛾+19440𝐴6𝛿2𝛽2𝛾17280𝐴5𝛿𝛽22𝛾+19440𝐴5𝛾𝛿2+5760𝐴5𝛿𝛾319440𝐴5𝛿2𝛾𝜎2𝐴cos(𝑥)67201+𝜎141080𝐴3𝛾𝛿𝜎2+540𝐴3𝛾𝛿+540𝐴3𝛿𝛾𝜎22180𝐴3𝛿𝛽21𝛾1080𝐴4𝛿𝛽2𝛾𝜎2720𝐴4𝛿𝛽1𝛽222160𝐴5𝛿2𝛽1𝛽2+1080𝐴4𝛾𝛽2𝛿+720𝐴3𝛿𝛽1𝜎2𝛽2+6480𝐴6𝛿2𝛽2𝛾2340𝐴5𝛿𝛽22𝛾6480𝐴5𝛿2𝛾𝜎2720𝐴3𝛿𝛽1𝛽2+14580𝐴7𝛿3𝛾+1440𝐴5𝛿𝛾3+6480𝐴5𝛾𝛿2+e899243𝐴8𝛿3𝛾1411+𝜎14cos7(𝑥)𝐴cos6(𝑥)67201+𝜎1490720𝐴6𝛿2𝛽2𝛾25515𝐴6𝛿3𝛽1𝐴67201+𝜎14720𝐴5𝛿𝛽22𝛾+45𝐴2𝛿𝛽1𝜎22+405𝐴6𝛿3𝛽1+720𝐴4𝛾𝛽2𝛿720𝐴4𝛿𝛽2𝛾𝜎2+90𝐴3𝛿𝛽1𝛽2+45𝐴4𝛿𝛽1𝛽22+2160𝐴6𝛿2𝛽2𝛾+270𝐴5𝛿2𝛽1𝛽2+45𝐴2𝛿𝛽190𝐴3𝛿𝛽1𝜎2𝛽2+270𝐴4𝛿2𝛽1270𝐴4𝛿2𝛽1𝜎290𝐴2𝛿𝛽1𝜎2+180𝐴4𝛿𝛽1𝛾2sin(𝑥)𝐴cos5(𝑥)67201+𝜎14204120𝐴8𝛿47200𝐴4𝛿𝛽1𝛾𝛽2+52245𝐴6𝛿3+2025𝛿2𝐴452245𝐴6𝛿3𝜎2+3600𝐴4𝛾2𝛿80055𝐴6𝛿2𝛽223600𝐴5𝛿𝛽32+3600𝛿𝛽2𝛾2𝐴5+52245𝐴7𝛿3𝛽2+82080𝛿2𝛾2𝐴6+2025𝐴4𝛿2𝜎22+3600𝐴4𝛿𝛽22𝜎24050𝐴5𝛿2𝛽2𝜎24050𝐴4𝛿2𝜎22025𝐴4𝛿2𝛽21+4050𝐴5𝛿2𝛽23600𝐴4𝛿𝛽223600𝐴4𝛿𝛾2𝜎2243𝐴8𝛿3𝛽2cos8(𝑥)1411+𝜎14𝐴cos4(𝑥)67201+𝜎1425920𝛿𝛽2𝛾2𝐴5+1440𝐴4𝛿𝛽22+166860𝐴7𝛿3𝛽2+33480𝐴5𝛿2𝛽21440𝐴4𝛿𝛽22𝜎2+1440𝐴3𝛾𝛽1𝛿720𝐴3𝛿𝛽21𝛽2+33480𝐴6𝛿2𝛽221440𝐴3𝛿𝛽1𝛾𝜎27920𝐴5𝛿𝛽32+1440𝐴4𝛿𝛽1𝛾𝛽2+27000𝐴5𝛿2𝛽1𝛾33480𝐴5𝛿2𝛽2𝜎21440𝐴3𝛿𝛽2𝜎2+720𝐴3𝛿𝛽2𝜎22+720𝛿𝛽2𝐴3𝐴cos3(𝑥)67201+𝜎142025𝐴4𝛿2𝛽21+6480𝐴4𝛿2𝜎2+5040𝐴4𝛿𝛾2𝜎2+135𝐴3𝛿𝛽21𝛽23240𝛿2𝐴4+7920𝐴4𝛿𝛽1𝛾𝛽270470𝐴8𝛿432805𝐴6𝛿34905𝐴4𝛿𝛽22𝜎2+135𝐴2𝛿𝛽21+270𝐴3𝛿𝛽2𝜎2+6480𝐴5𝛿2𝛽2𝜎2135𝐴3𝛿𝛽2𝜎22135𝐴2𝛿𝜎22+135𝐴2𝜎2𝛿+45𝐴2𝛿𝜎3245𝐴2𝛿32805𝐴7𝛿3𝛽245900𝛿2𝛾2𝐴6+32805𝐴6𝛿3𝜎2+37800𝐴6𝛿2𝛽22135𝛿𝛽2𝐴3+4995𝐴5𝛿𝛽32135𝐴2𝛿𝛽21𝜎26480𝐴5𝛿2𝛽25040𝛿𝛽2𝛾2𝐴5+4905𝐴4𝛿𝛽225040𝐴4𝛾2𝛿3240𝐴4𝛿2𝜎226561𝐴9𝛿4cos9(𝑥)44811+𝜎14𝐴cos2(𝑥)67201+𝜎14540𝐴3𝛿𝛽21𝛽2+1980𝐴5𝛿𝛽329180𝐴5𝛿2𝛽1𝛾900𝛿𝛽2𝐴3+1800𝐴4𝛿𝛽22𝜎215120𝐴5𝛿2𝛽210080𝛿𝛽2𝛾2𝐴515120𝐴6𝛿2𝛽22+1440𝐴3𝛿𝛽1𝛾𝜎21440𝐴3𝛾𝛽1𝛿37260𝐴7𝛿3𝛽21440𝐴4𝛿𝛽1𝛾𝛽2+15120𝐴5𝛿2𝛽2𝜎2+1800𝐴3𝛿𝛽2𝜎2900𝐴3𝛿𝛽2𝜎221800𝐴4𝛿𝛽22𝐴67201+𝜎14180𝐴4𝛿𝛽1𝛾𝛽2+360𝐴4𝛿𝛽22+180𝛿𝛽2𝐴3360𝐴4𝛿𝛽22𝜎2+1080𝐴6𝛿2𝛽22+1080𝐴5𝛿2𝛽2+1620𝐴7𝛿3𝛽2180𝐴3𝛿𝛽1𝛾𝜎2+180𝐴3𝛾𝛽1𝛿+180𝐴5𝛿𝛽32+540𝐴5𝛿2𝛽1𝛾1080𝐴5𝛿2𝛽2𝜎2360𝐴3𝛿𝛽2𝜎2+180𝐴3𝛿𝛽2𝜎22+720𝛿𝛽2𝛾2𝐴5𝐴cos7(𝑥)67201+𝜎1445360𝐴6𝛿2𝛽22+25515𝐴6𝛿3𝜎225515𝐴6𝛿345360𝛿2𝛾2𝐴6240570𝐴8𝛿425515𝐴7𝛿3𝛽2𝐴cos(𝑥)67201+𝜎14270𝐴4𝛿2𝛽212430𝐴4𝛿2𝜎21620𝐴4𝛿𝛾2𝜎290𝐴3𝛿𝛽21𝛽2+1215𝛿2𝐴41440𝐴4𝛿𝛽1𝛾𝛽2+8505𝐴8𝛿4+6075𝐴6𝛿3+1305𝐴4𝛿𝛽22𝜎290𝐴2𝛿𝛽21270𝐴3𝛿𝛽2𝜎22430𝐴5𝛿2𝛽2𝜎2+135𝐴3𝛿𝛽2𝜎22+135𝐴2𝛿𝜎22135𝐴2𝜎2𝛿45𝐴2𝛿𝜎32+45𝐴2𝛿+6075𝐴7𝛿3𝛽2+8100𝛿2𝛾2𝐴66075𝐴6𝛿3𝜎23105𝐴6𝛿2𝛽22+135𝛿𝛽2𝐴31395𝐴5𝛿𝛽32+90𝐴2𝛿𝛽21𝜎2+2430𝐴5𝛿2𝛽2+1620𝛿𝛽2𝛾2𝐴51305𝐴4𝛿𝛽22+1620𝐴4𝛾2𝛿+1215𝐴4𝛿2𝜎22𝑡8+𝐴cos6(𝑥)67201+𝜎1472576𝐴5𝛿2𝛽2𝜎1+72576𝐴5𝛿2𝛽2+𝐴cos5(𝑥)67201+𝜎1499792𝐴5𝛾𝛿2𝜎199792𝐴5𝛾𝛿2𝐴cos2(𝑥)67201+𝜎141008𝐴3𝛿𝛽1𝛽2+1344𝐴2𝛾2𝛽11344𝐴2𝛾𝛽2𝜎2𝜎11008𝐴2𝛿𝛽1𝜎2+1344𝐴3𝛾𝛽22+1344𝐴2𝛾𝛽2𝜎1+1344𝐴3𝛾𝛽22𝜎1+1344𝐴2𝛾𝛽21008𝐴2𝛿𝛽1𝜎1𝜎2+1344𝐴2𝛾2𝛽1𝜎1+12096𝐴4𝛿2𝛽1+1008𝐴3𝛿𝛽1𝜎1𝛽2+12096𝐴4𝛿2𝛽1𝜎11344𝐴2𝛾𝛽2𝜎2+40320𝐴4𝛾𝛽2𝛿𝜎1+40320𝐴4𝛾𝛽2𝛿+1008𝐴2𝛿𝛽1+1008𝐴2𝛿𝛽1𝜎1𝐴67201+𝜎14448𝐴2𝛾𝛽2+448𝐴2𝛾𝛽2𝜎2𝜎1448𝐴2𝛾𝛽2𝜎1448𝐴3𝛾𝛽22𝜎1+448𝐴2𝛾𝛽2𝜎2+336𝐴2𝛿𝛽1𝜎21008𝐴4𝛿2𝛽1336𝐴3𝛿𝛽1𝛽2336𝐴2𝛿𝛽1336𝐴3𝛿𝛽1𝜎1𝛽2+336𝐴2𝛿𝛽1𝜎1𝜎2336𝐴2𝛿𝛽1𝜎14032𝐴4𝛾𝛽2𝛿224𝐴2𝛾2𝛽1448𝐴3𝛾𝛽224032𝐴4𝛾𝛽2𝛿𝜎1224𝐴2𝛾2𝛽1𝜎11008𝐴4𝛿2𝛽1𝜎1𝐴cos(𝑥)67201+𝜎142688𝐴3𝛿𝛽1𝛽2+1680𝐴3𝛾𝛽22𝜎11792𝐴3𝛾3+2688𝐴3𝛿𝛽1𝜎1𝛽2+6720𝐴3𝛾𝛿𝜎2𝜎16720𝐴3𝛾𝛿𝜎1+6720𝐴3𝛾𝛿𝜎231248𝐴5𝛾𝛿2224𝐴2𝛾𝛽2𝜎1+224𝐴2𝛾𝛽2𝜎2𝜎1+1680𝐴3𝛾𝛽22+224𝐴2𝛾𝛽2𝜎21792𝐴3𝛾3𝜎16720𝐴3𝛾𝛿224𝐴2𝛾𝛽2112𝐴𝛾+112𝐴𝛾𝛽21+112𝐴𝛾𝛽21𝜎1112𝐴𝛾𝜎22𝜎1+224𝐴𝛾𝜎2𝜎1112𝐴𝛾𝜎22+224𝐴𝛾𝜎2112𝐴𝛾𝜎16720𝐴4𝛾𝛽2𝛿𝜎131248𝐴5𝛾𝛿2𝜎16720𝐴4𝛾𝛽2𝛿𝐴cos3(𝑥)6720/1+𝜎143584𝐴3𝛾𝛽22𝜎1+9408𝐴4𝛾𝛽2𝛿𝜎1+118944𝐴5𝛾𝛿2+9408𝐴4𝛾𝛽2𝛿+3584𝐴3𝛾3𝜎1+9408𝐴3𝛾𝛿+9408𝐴3𝛾𝛿𝜎1+118944𝐴5𝛾𝛿2𝜎13584𝐴3𝛾𝛽225376𝐴3𝛿𝛽1𝛽2+3584𝐴3𝛾39408𝐴3𝛾𝛿𝜎25376𝐴3𝛿𝛽1𝜎1𝛽29408𝐴3𝛾𝛿𝜎2𝜎1𝐴cos4(𝑥)67201+𝜎1415120𝐴4𝛿2𝛽147040𝐴4𝛾𝛽2𝛿15120𝐴4𝛿2𝛽1𝜎147040𝐴4𝛾𝛽2𝛿𝜎1sin(𝑥)𝐴cos7(𝑥)67201+𝜎1495256𝐴6𝛿3𝜎1+95256𝐴6𝛿3𝐴cos3(𝑥)67201+𝜎1424192𝐴4𝛿2𝜎224192𝐴4𝛿2𝜎1𝜎2+24192𝛿2𝐴41344𝐴2𝛾𝛽1𝛽2𝜎1+122472𝐴6𝛿3+1008𝐴3𝛽2𝛿𝜎11344𝐴2𝛾𝛽1𝛽218312𝐴4𝛿𝛽22𝜎1+1344𝐴3𝛾2𝛽2𝜎1504𝐴2𝛿𝛽21+1344𝐴2𝛾2𝜎11008𝐴3𝛿𝛽2𝜎2+24192𝐴5𝛿2𝛽2𝜎1+504𝐴2𝛿𝜎1+24192𝐴4𝛿2𝜎1+54024𝐴4𝛾2𝛿𝜎11344𝐴2𝛾2𝜎2+504𝐴2𝛿𝜎221008𝐴2𝜎2𝛿+1008𝐴2𝜎2𝛿𝜎1+504𝐴2𝛿1008𝐴3𝛿𝛽2𝜎1𝜎21344𝐴2𝛾2𝜎2𝜎1122472𝐴6𝛿3𝜎1+1008𝛿𝛽2𝐴3+1344𝐴2𝛾2+504𝐴2𝛿𝜎1𝜎22504𝐴2𝛿𝛽21𝜎1+24192𝐴5𝛿2𝛽218312𝐴4𝛿𝛽22+45024𝐴4𝛾2𝛿+1344𝐴3𝛾2𝛽2𝐴cos5(𝑥)67201+𝜎14195048𝐴6𝛿3+15120𝐴4𝛿2𝜎1𝜎2+15120𝐴4𝛿2𝜎2195048𝐴6𝛿3𝜎115120𝐴5𝛿2𝛽2+13440𝐴4𝛿𝛽2233600𝐴4𝛾2𝛿𝜎133600𝐴4𝛾2𝛿15120𝛿2𝐴415120𝐴4𝛿2𝜎115120𝐴5𝛿2𝛽2𝜎1+13440𝐴4𝛿𝛽22𝜎1𝐴67201+𝜎14112𝐴𝛾𝛽1𝜎11344𝛿𝛽2𝐴3112𝐴2𝛾𝛽1𝛽21008𝐴3𝛾𝛽1𝛿𝜎1+112𝐴𝛾𝛽1𝜎2𝜎1+1344𝐴3𝛿𝛽2𝜎1𝜎2896𝐴3𝛾2𝛽2112𝐴2𝛾𝛽1𝛽2𝜎11344𝐴3𝛽2𝛿𝜎11344𝐴4𝛿𝛽22𝜎11008𝐴3𝛾𝛽1𝛿112𝐴𝛾𝛽1+112𝐴𝛾𝛽1𝜎2896𝐴3𝛾2𝛽2𝜎14032𝐴5𝛿2𝛽21344𝐴4𝛿𝛽22+1344𝐴3𝛿𝛽2𝜎24032𝐴5𝛿2𝛽2𝜎1𝐴cos4(𝑥)67201+𝜎147168𝐴3𝛾2𝛽25376𝐴4𝛿𝛽22𝜎19408𝐴3𝛾𝛽1𝛿9408𝐴3𝛾𝛽1𝛿𝜎15376𝐴3𝛽2𝛿𝜎17168𝐴3𝛾2𝛽2𝜎15376𝛿𝛽2𝐴3124992𝐴5𝛿2𝛽25376𝐴4𝛿𝛽22+5376𝐴3𝛿𝛽2𝜎2+5376𝐴3𝛿𝛽2𝜎1𝜎2124992𝐴5𝛿2𝛽2𝜎1𝐴cos(𝑥)67201+𝜎149072𝐴4𝛿2𝜎2+9072𝐴4𝛿2𝜎1𝜎29072𝛿2𝐴4+896𝐴2𝛾𝛽1𝛽2𝜎122680𝐴6𝛿31008𝐴3𝛽2𝛿𝜎1+896𝐴2𝛾𝛽1𝛽2+4872𝐴4𝛿𝛽22𝜎11120𝐴3𝛾2𝛽2𝜎1+336𝐴2𝛿𝛽211120𝐴2𝛾2𝜎1+1008𝐴3𝛿𝛽2𝜎29072𝐴5𝛿2𝛽2𝜎1504𝐴2𝛿𝜎19072𝐴4𝛿2𝜎113440𝐴4𝛾2𝛿𝜎1+1120𝐴2𝛾2𝜎2504𝐴2𝛿𝜎22+1008𝐴2𝜎2𝛿+1008𝐴2𝜎2𝛿𝜎1504𝐴2𝛿+1008𝐴3𝛿𝛽2𝜎1𝜎2+1120𝐴2𝛾2𝜎2𝜎122680𝐴6𝛿3𝜎11008𝛿𝛽2𝐴31120𝐴2𝛾2504𝐴2𝛿𝜎1𝜎22+336𝐴2𝛿𝛽21𝜎19072𝐴5𝛿2𝛽2+4872𝐴4𝛿𝛽2213440𝐴4𝛾2𝛿1120𝐴3𝛾2𝛽2𝐴cos2(𝑥)67201+𝜎14224𝐴𝛾𝛽1𝜎2𝜎1+224𝐴𝛾𝛽1𝜎1+224𝐴2𝛾𝛽1𝛽2+56448𝐴5𝛿2𝛽2+6720𝐴4𝛿𝛽22+9072𝐴3𝛾𝛽1𝛿𝜎1+224𝐴𝛾𝛽1+6720𝛿𝛽2𝐴3+9072𝐴3𝛾𝛽1𝛿+224𝐴2𝛾𝛽1𝛽2𝜎1+7168𝐴3𝛾2𝛽2+56448𝐴5𝛿2𝛽2𝜎1+6720𝐴3𝛽2𝛿𝜎1+7168𝐴3𝛾2𝛽2𝜎16720𝐴3𝛿𝛽2𝜎1𝜎2224𝐴𝛾𝛽1𝜎2+6720𝐴4𝛿𝛽22𝜎16720𝐴3𝛿𝛽2𝜎2𝑡6+𝐴cos3(𝑥)67201+𝜎141008𝐴2𝛽2𝛽1𝛼𝜎214032𝐴2𝛽22𝜎11008𝐴2𝛽2𝛽1𝛼2016𝐴2𝛽2𝛽1𝛼𝜎12016𝐴2𝛽222016𝐴2𝛽22𝜎21𝐴cos(𝑥)67201+𝜎141344𝐴2𝛽22𝜎21+672𝐴2𝛼𝛽2𝛽1+1344𝐴2𝛽22+2688𝐴2𝛽22𝜎1+1344𝐴2𝛽2𝛽1𝛼𝜎1+672𝐴2𝛽2𝛽1𝛼𝜎21𝐴67201+𝜎14504𝛿𝛽2𝐴3336𝐴2𝛽22𝜎1168𝐴𝛽2𝜎21+336𝐴𝛽2𝜎2𝜎1+168𝐴𝛽2𝜎21008𝐴3𝛽2𝛿𝜎1168𝛽2𝐴504𝐴3𝛽2𝛿𝜎21168𝐴2𝛽22672𝐴3𝛽2𝛾𝛼𝜎21168𝐴2𝛽22𝜎21336𝐴𝛽2𝜎1+168𝐴𝛽2𝜎2𝜎21672𝐴3𝛽2𝛾𝛼1344𝐴3𝛽2𝛾𝛼𝜎1𝐴cos2(𝑥)67201+𝜎145544𝐴3𝛽2𝛿𝜎21+5376𝐴3𝛽2𝛾𝛼𝜎21672𝐴𝛽2𝜎2𝜎1336𝐴𝛽2𝜎2+336𝛽2𝐴+672𝐴2𝛽22𝜎1+336𝐴2𝛽22+5376𝐴3𝛽2𝛾𝛼+672𝐴𝛽2𝜎1336𝐴𝛽2𝜎2𝜎21+336𝐴𝛽2𝜎21+10752𝐴3𝛽2𝛾𝛼𝜎1+336