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

Authors: Khaled A. Gepreel; S. M. Abo-Dahab; T. A. Nofal
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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 [1–4]. 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 [5–7].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 [8–20] 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. [25–27] 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]:(2.1)A(u)-f(r)=0,r∈Λ,
with the boundary conditions:(2.2)B(u,∂u∂n)=0,r∈Γ,
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:(2.3)L(u)+N(u)-f(r)=0.
By the homotopy technique, we construct a homotopy V(r,p): Λ×[0,1]→R which satisfies(2.4)H(V,p)=(1-p)[L(V)-L(u0)]+p[A(V)-f(r)]=0,r∈Λ,
or(2.5)H(V,p)=L(V)-L(u0)+pL(u0)+p(N(V)-f(r))=0,r∈Λ,
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 have(2.6)H(V,0)=L(V)-L(u0)=0,H(V,1)=A(V)-f(r)=0.
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:(2.7)V=V0+pV1+p2V2+⋯.
On setting p=1 results in the approximate solution of (2.3), we have(2.8)u=lim⁡p→1⁡V=V0+V1+V2+⋯.
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 ofN(V) with respect to Vmust be small because the parameter may be relatively large, that is, p→1.(2)
The norm ofL-1(∂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:(3.1)(1+σ1)utt+Ωut-uxx(1-σ2+2γux+3δux2)-β1θx-β2(θux)x=0,(θ-aux-12bux2)t-[(1+αux)θx]x=0,
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 conditions(3.2)u(x,0)=θ(x,0)=A(1-cos⁡⁡(x)),ut(x,0)=θt(x,0)=0,
where A is an arbitrary constant and the boundary conditions(3.3)u(0,t)=θ(0,t)=0,ut(0,t)=θt(0,t)=0.
To investigate the traveling wave solution of (3.1), we first construct a homotopy perturbation method as follows:(3.4)(1-p)[(1+σ1)(Vtt-V0tt)]+p[(1+σ1)Vtt+ΩVt-Vxx(1-σ2+2γVx+3δVx2)-β1Θx-β2(ΘVx)x]=0,(1-p)(Θt-Θ0t)+p[Θt-aVxt-bVxVxt-Θxx-αVxxΘx-αVxΘxx]=0,
where the initial approximations take the following form:(3.5)V0(x,t)=u0(x,t)=u(x,0)=A(1-cos⁡⁡(x)),Θ0(x,t)=θ0(x,t)=θ(x,0)=A(1-cos⁡⁡(x)).
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:(3.6)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)+⋯,
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 have(3.7)(1+σ1)V0,tt+(σ1V1,tt-3δV0,x2V0,xx-β2V0,xxΘ0-2γV0,xV0,xx+σ2V0,xx-β1Θ0,x-V0,xx-β2V0,xΘ0,x+ΩV0,t+V1,ttV0,x2)p+(V0,x2σ1V2,tt-2γV1,xV0,xx+V2,tt-2γV0,xV1,xx-6δV0,xV1,xV0,xx-V1,xx-β2V1,xxΘ0-β1Θ1,x-β2V0,xΘ1,x-β2V1,xΘ0,x-β2V0,xxΘ1+ΩV1,t-3δV0,x2V1,xx+σ2V1,xx)p2+(V3,tt-β2V2,xΘ0,x-β2V2,xxΘ0-2γV2,xV0,xx-2γV1,xV1,xx+σ1V3,tt-β1Θ2,x-6δV0,xV2,xV0,xx-3δV1,x2V0,xx+σ2V2,xx+ΩV2,t-V2,xx-3δV0,x2V2,xx-2γV0,xV2,xx-β2V1,xxΘ1-β2V0,xxΘ2-β2V0,xΘ2,x-6δV0,xV1,xV1,xx-β2V1,xΘ1,x)p3+⋯=0,Θ0,t+(-aV0,xt-αV0,xΘ0,xx+Θ1,t-αV0,xxΘ0,x-2bV0,xV0,xt-Θ0,xx)p+(Θ2,t-Θ1,xx-aV1,xt-αV1,xxΘ0,x-2bV1,xV0,xt-αV0,xΘ1,xx-αV1,xΘ0,xx-2bV0,xV1,xt-αV0,xxΘ1,x)p2×(-αV1,xxΘ1,x-αV2,xxΘ0,x-2bV2,xV0,xt-αV0,xxΘ2,x-2bV1,xV1,xt-aV2,xt+Θ3,t-2bV0,xV2,xt-αV0,xΘ2,xx-αV1,xΘ1,xx-αV2,xΘ0,xx-Θ2,xx)p3+⋯=0.
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):(3.8)σ1V1,tt-3δV0,x2V0,xx-β2V0,xxΘ0-2γV0,xV0,xx+σ2V0,xx-β1Θ0,x-V0,xx-β2V0,xΘ0,x+ΩV0,t+V1,tt=0,σ1V2,tt-2γV1,xV0,xx+V2,tt-2γV0,xV1,xx-6δV0,xV1,xV0,xx-V1,xx-β2V1,xxΘ0-β1Θ1,x-β2V0,xΘ1,x-β2V1,xΘ0,x-β2V0,xxΘ1+ΩV1,t-3δV0,x2V1,xx+σ2V1,xx=0,V3,tt-β2V2,xΘ0,x-β2V2,xxΘ0-2γV2,xV0,xx-2γV1,xV1,xx+σ1V3,tt-β1Θ2,x-6δV0,xV2,xV0,xx-3δV1,x2V0,xx+σ2V2,xx+ΩV2,t-V2,xx-3δV0,x2V2,xx-2γV0,xV2,xx-β2V1,xxΘ1-β2V0,xxΘ2-β2V0,xΘ2,x-6δV0,xV1,xV1,xx-β2V1,xΘ1,x=0,-aV0,xt-αV0,xΘ0,xx+Θ1,t-αV0,xxΘ0,x-2bV0,xV0,xt-Θ0,xx=0,Θ2,t-Θ1,xx-aV1,xt-αV1,xxΘ0,x-2bV1,xV0,xt-αV0,xΘ1,xx-αV1,xΘ0,xx-2bV0,xV1,xt-αV0,xxΘ1,x=0,-αV1,xxΘ1,x-αV2,xxΘ0,x-2bV2,xV0,xt-αV0,xxΘ2,x-2bV1,xV1,xt-aV2,xt+Θ3,t-2bV0,xV2,xt-αV0,xΘ2,xx-αV1,xΘ1,xx-αV2,xΘ0,xx-Θ2,xx=0.
Consequently, we deduce after some calculations the following results:(3.9)u=lim⁡p→1⁡V=V0+V1+V2+⋯,θ=lim⁡p→1⁡Θ=Θ0+Θ1+Θ2+⋯,
where (3.10)V0=A(1-cos⁡⁡x),V1=t22(4+4σ1)[4γA2sin⁡⁡2x+3δA3cos⁡⁡x-3δA3cos⁡⁡3x+4β2A2cos⁡⁡x-4β2A2cos⁡⁡2x4σ2Acos⁡⁡x+4β1Asin⁡⁡x+4Acos⁡⁡x],V2=t432(σ1+1)2{(-83β22A3-83β2A2-83γ2A3-4δA3+4A3σ2δ+83σ2A-92δ2A5-43Aσ22-43A+83A2σ2β2-4β2A4δ)cos⁡⁡x-(43β1A+δA3β1+43γA3β2-43Aσ2β1+43β2A2β1)sin⁡⁡x+(-8γA2+8A2σ2γ-8β2A3γ+43β2A2β1-16δA4γ)sin⁡⁡2x+(12γA3β2+3δA3β1)sin⁡⁡3x+20δA4γsin⁡⁡4x+(-203A2σ2β2+203β22A3+83γA2β1+203β2A2+6δA4β2)cos⁡⁡2x+(634δ2A5-4β22A3+12δA3+12β2A4δ+8γ2A3-12A3σ2δ)cos⁡⁡3x-14β2A4δcos⁡4x-454δ2A5cos⁡5x}+t332(σ1+1)2{(-163ΩAβ1-163β1Aσ1-83β2ασ1A3-163β1A-83β2A3α)sin⁡⁡x-163ΩA2γsin⁡⁡2x+(8β2A3ασ1+8β2A3α)sin⁡⁡3x+(163ΩAσ2-4ΩA3δ-163ΩA2β2-163ΩA)cos⁡⁡(x)+(163β2A2+323β1A2ασ1+163ΩA2β2+163β2A2σ1+323β1A2α)cos⁡⁡2x+4ΩA3δcos⁡⁡3x(-163ΩAβ1-163β1Aσ1-83β2ασ1A3-163β1A-83β2A3α)},Θ0=A(1-cos⁡⁡(x)),Θ1=A2tαsin⁡⁡(2x)+Acos⁡⁡(x)t,Θ2=A2(1+σ1)(t33(12αA2γcos⁡3(x)+2αAcos⁡2(x)β1+36αA3δsin⁡⁡(x)cos⁡3(x)+12αA2β2sin⁡(x)cos⁡2(x)+2αAσ2sin⁡(x)cos⁡(x)-24αA3δsin⁡(x)cos⁡(x)-2αA2β2sin⁡(x)cos⁡(x)-2αAsin⁡(x)cos⁡(x)-αAβ1-10αγA2cos⁡⁡(x)-4αβ2A2sin⁡⁡(x)t33)+t22(t33-2cos⁡⁡(x)+18aδA2sin⁡(x)cos⁡2(x)+16bA2γsin⁡(x)cos⁡2(x)+8aβ2Asin⁡(x)cos⁡(x)+2aσ2sin⁡(x)+2aβ1cos⁡(x)+24α2A2σ1cos⁡3(x)-36bA3δcos⁡4(x)-16bA2β2cos⁡3(x)+48bA3δcos⁡2(x)+4bA2β2cos⁡2(x)+8cos⁡2(x)aγA-4cos⁡⁡(x)2bAσ2-20α2A2cos⁡⁡(x)+24cos⁡⁡(x)3α2A2+4bAcos⁡2(x)-4bA-6aδA2sin⁡(x)-20αAsin⁡(x)cos⁡(x)-2aβ2Asin⁡(x)-4bA2β2-12bA3δ+4bAσ2-8bA2sin⁡(x)γ+16bA2β2cos⁡(x)-20α2A2cos⁡(x)σ1-4aγA-2cos⁡(x)σ1-2asin⁡(x)+4bAβ1sin⁡(x)cos⁡(x)-20αAsin⁡(x)cos⁡(x)σ1A2)t33).
Now we make calculations for the results obtained by the homotopy perturbation method using the Maple software package with the following arbitrary constants:(3.11)a=0.5,A=0.001,b=0.5,α=1,β1=β2=0.05,γ=1,δ=0.8.
The results obtained in (3.9) are displayed graphically in Figures 1–4.Variations of the displacementu and temperature θ for various values of the axis x and time t whenΩ=0.1, σ1=0.2, σ2=0.1.
(a)(b)Variations of the displacementu and temperature θ for various values of the axis x and rotation Ω when t=0.1, σ1=0.2, σ2=0.1.
(a)(b)Variations of the displacementu and temperature θ for various values of the axis x and magnetic field σ1 whent=0.1, Ω=0.1, σ2=0.1.
(a)(b)Variations of the displacementu and temperature θ for various values of the axis x and magnetic field σ2 when t=0.1, Ω=0.1, σ1=0.1.
(a)(b)


### 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 displacementu and temperature θ take the following forms. See Figures 9 and 10.Variations of the displacementu and temperature θ for various values of the axis x and time t when Ω=0.1, σ1=0.2, σ2=0.1.
(a)(b)Variations of the displacementu and temperature θ for various values of the axis x and rotation Ω when t=0.1, σ1=0.2, σ2=0.1.
(a)(b)Variations of the displacementu and temperature θ for various values of the axis x and magnetic field σ1 when t=0.1, Ω=0.1, σ2=0.1.
(a)(b)Variations of the displacementu and temperature θ for various values of the axis x and magnetic field σ2 when t=0.1, Ω=0.1, σ1=0.1.
(a)(b)Variations of the displacementu 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.
(a)(b)Variations of the displacementu 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.
(a)(b)


## 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 displacementu and temperature θ take the following forms. See Figures 9 and 10.Variations of the displacementu and temperature θ for various values of the axis x and time t when Ω=0.1, σ1=0.2, σ2=0.1.
(a)(b)Variations of the displacementu and temperature θ for various values of the axis x and rotation Ω when t=0.1, σ1=0.2, σ2=0.1.
(a)(b)Variations of the displacementu and temperature θ for various values of the axis x and magnetic field σ1 when t=0.1, Ω=0.1, σ2=0.1.
(a)(b)Variations of the displacementu and temperature θ for various values of the axis x and magnetic field σ2 when t=0.1, Ω=0.1, σ1=0.1.
(a)(b)Variations of the displacementu 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.
(a)(b)Variations of the displacementu 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.
(a)(b)


## 4. Basic Idea of Variational Iteration Method

Consider the following nonhomogeneous nonlinear system of partial differential equations:(4.1)L1u(x,t)+N1(u(x,t),θ(x,t))=f(x,t),(4.2)L2θ(x,t)+N2(u(x,t),θ(x,t))=g(x,t),
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:(4.3)un+1(x,t)=un(x,t)+∫0tλ1(τ)[L1un(x,τ)+N1(ũn(x,τ),θ̃n(x,τ))-f(x,τ)]dτ,(4.4)θn+1(x,t)=θn(x,t)+∫0tλ2(τ)[L2θn(x,τ)+N2(ũn(x,τ),θ̃n(x,τ))-g(x,τ)]dτ,
where λ1 and λ2are general Lagrange multipliers, which can be identified optimally via variational theory [8–20]. 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, ũn and θ̃n 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:(5.1)un+1(x,t)=un(x,t)+∫0tλ1(τ)[β2(θ̃n(x,τ)ũn,x(x,τ))x(1+σ1)un,tt(x,τ)+Ωũn,t(x,τ)-ũn,xx(1-σ2+2γũn,x(x,τ)+3δũn,x2(x,τ))-β1θ̃n,x(x,τ)-β2(θ̃n(x,τ)ũn,x(x,τ))x]dτ,θn+1(x,t)=θn(x,t)+∫0tλ2(τ)[β2(θ̃n(x,τ)ũn,x(x,τ))xθn,t(x,τ)-aũn,xt(x,τ)-bũn,x(x,τ)ũn,xt(x,τ)-θ̃n,xx(x,τ)-αũn,xx(x,τ)θ̃n,x(x,τ)-αũn,x(x,τ)θ̃n,xx(x,τ)]dτ,
where ũn and θ̃n are considered as a restricted variation, that is, δũn+1=0 and δθ̃n+1=0. Consequently, the general Lagrange multipliers λ1 and λ2 take the following form:(5.2)λ1(τ)=τ-t1+σ1,λ2(τ)=-1.
By the substitution of the identified Lagrange multipliers (5.2) into (5.1), we have the following iteration relations:(5.3)un+1(x,t)=un(x,t)+∫0tτ-t1+σ1[β2(θ̃n(x,τ)ũn,x(x,τ))x(1+σ1)un,tt(x,τ)+Ωun,t(x,τ)-un,xx(1-σ2+2γun,x(x,τ)+3δun,x2(x,τ))-β1θn,x(x,τ)-β2(θn(x,τ)un,x(x,τ))x]dτ,θn+1(x,t)=θn(x,t)-∫0t[β2(θ̃n(x,τ)ũn,x(x,τ))xθn(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,n≥0.
With help of Maple or Mathematica, we get the following results:(5.4)u0=θ0=A(1-cos⁡x),u1=-t22(1+σ1)[-β1sin⁡x-2Aγcos⁡xsin⁡(x)+2β2Acos⁡2x-cos⁡x-3δA2cos⁡x-β2A+σ2cos⁡x-β2Acos⁡x+3δA2cos⁡3x]+A(1-cos⁡x),θ1=A[cos⁡x+2αAcos⁡xsin⁡x]t+A(1-cos⁡x),u2={-Acos⁡6(x)6720(1+σ1)4(-19440A5δ2β1γ+5760A5δβ23-247860A7δ3β2-19440A6δ2β22-19440A5δ2β2-17280δβ2γ2A5+19440A5δ2β2σ2)+[-Acos⁡2(x)6720(1+σ1)4(-6480A5δβ22γ+2025A4δβ1β22+270A3δβ1σ2β2+3240A4δ2β1σ2-38880A6δ2β2γ-8505A6δ3β1-135A2δβ1σ22-6480A4γβ2δ-135A2δβ1+6480A4δβ2γσ2-3240A5δ2β1β2+270A2δβ1σ2-270A3δβ1β2-3240A4δ2β1-2160A4δβ1γ2+45A2δβ13)-Acos⁡4(x)6720(1+σ1)4(7200A4γβ2δ+118800A6δ2β2γ-4050A4δ2β1σ2+4050A4δ2β1+4050A5δ2β1β2+3600A4δβ1γ2-3600A4δβ1β22+7200A5δβ22γ+30375A6δ3β1-7200A4δβ2γσ2)-Acos⁡3(x)6720(1+σ1)4(17280A5δ2β1β2-1440A4γβ2δ-93960A7δ3γ+16560A5δβ22γ+1440A4δβ2γσ2-23760A6δ2β2γ+1440A4δβ1β22-720A3γδ+1440A3δβ1β2+23760A5δ2γσ2-720A3δγσ22-5760A5δγ3-23760A5γδ2-1440A3δβ1σ2β2+720A3δβ12γ+1440A3γδσ2)-Acos⁡5(x)6720(1+σ1)4(-19440A5δ2β1β2+189540A7δ3γ+19440A6δ2β2γ-17280A5δβ22γ+19440A5γδ2+5760A5δγ3-19440A5δ2γσ2)-Acos⁡(x)6720(1+σ1)4(β22-1080A3γδσ2+540A3γδ+540A3δγσ22-180A3δβ12γ-1080A4δβ2γσ2-720A4δβ1β22-2160A5δ2β1β2+1080A4γβ2δ+720A3δβ1σ2β2+6480A6δ2β2γ-2340A5δβ22γ-6480A5δ2γσ2-720A3δβ1β2+14580A7δ3γ+1440A5δγ3+6480A5γδ2β22)e899+243A8δ3γ141(1+σ1)4cos⁡7(x)-Acos⁡6(x)6720(1+σ1)4(-90720A6δ2β2γ-25515A6δ3β1)-A6720(1+σ1)4(720A5δβ22γ+45A2δβ1σ22+405A6δ3β1+720A4γβ2δ-720A4δβ2γσ2+90A3δβ1β2+45A4δβ1β22+2160A6δ2β2γ+270A5δ2β1β2+45A2δβ1-90A3δβ1σ2β2+270A4δ2β1-270A4δ2β1σ2-90A2δβ1σ2+180A4δβ1γ2)Acos⁡(x)6720(1+σ1)4]sin⁡(x)-Acos⁡5(x)6720(1+σ1)4(β22204120A8δ4-7200A4δβ1γβ2+52245A6δ3+2025δ2A4-52245A6δ3σ2+3600A4γ2δ-80055A6δ2β22-3600A5δβ23+3600δβ2γ2A5+52245A7δ3β2+82080δ2γ2A6+2025A4δ2σ22+3600A4δβ22σ2-4050A5δ2β2σ2-4050A4δ2σ2-2025A4δ2β12+4050A5δ2β2-3600A4δβ22-3600A4δγ2σ2β22)-243A8δ3β2cos⁡8(x)141(1+σ1)4-Acos⁡4(x)6720(1+σ1)4(β2225920δβ2γ2A5+1440A4δβ22+166860A7δ3β2+33480A5δ2β2-1440A4δβ22σ2+1440A3γβ1δ-720A3δβ12β2+33480A6δ2β22-1440A3δβ1γσ2-7920A5δβ23+1440A4δβ1γβ2+27000A5δ2β1γ-33480A5δ2β2σ2-1440A3δβ2σ2+720A3δβ2σ22+720δβ2A3β22)-Acos⁡3(x)6720(1+σ1)4(β222025A4δ2β12+6480A4δ2σ2+5040A4δγ2σ2+135A3δβ12β2-3240δ2A4+7920A4δβ1γβ2-70470A8δ4-32805A6δ3-4905A4δβ22σ2+135A2δβ12+270A3δβ2σ2+6480A5δ2β2σ2-135A3δβ2σ22-135A2δσ22+135A2σ2δ+45A2δσ23-45A2δ-32805A7δ3β2-45900δ2γ2A6+32805A6δ3σ2+37800A6δ2β22-135δβ2A3+4995A5δβ23-135A2δβ12σ2-6480A5δ2β2-5040δβ2γ2A5+4905A4δβ22-5040A4γ2δ-3240A4δ2σ22β22)-6561A9δ4cos⁡9(x)4481(1+σ1)4-Acos⁡2(x)6720(1+σ1)4(β22540A3δβ12β2+1980A5δβ23-9180A5δ2β1γ-900δβ2A3+1800A4δβ22σ2-15120A5δ2β2-10080δβ2γ2A5-15120A6δ2β22+1440A3δβ1γσ2-1440A3γβ1δ-37260A7δ3β2-1440A4δβ1γβ2+15120A5δ2β2σ2+1800A3δβ2σ2-900A3δβ2σ22-1800A4δβ22β22)-A6720(1+σ1)4(β22180A4δβ1γβ2+360A4δβ22+180δβ2A3-360A4δβ22σ2+1080A6δ2β22+1080A5δ2β2+1620A7δ3β2-180A3δβ1γσ2+180A3γβ1δ+180A5δβ23+540A5δ2β1γ-1080A5δ2β2σ2-360A3δβ2σ2+180A3δβ2σ22+720δβ2γ2A5β22)-Acos⁡7(x)6720(1+σ1)4(β2245360A6δ2β22+25515A6δ3σ2-25515A6δ3-45360δ2γ2A6-240570A8δ4-25515A7δ3β2β22)-Acos⁡(x)6720(1+σ1)4(β22-270A4δ2β12-2430A4δ2σ2-1620A4δγ2σ2-90A3δβ12β2+1215δ2A4-1440A4δβ1γβ2+8505A8δ4+6075A6δ3+1305A4δβ22σ2-90A2δβ12-270A3δβ2σ2-2430A5δ2β2σ2+135A3δβ2σ22+135A2δσ22-135A2σ2δ-45A2δσ23+45A2δ+6075A7δ3β2+8100δ2γ2A6-6075A6δ3σ2-3105A6δ2β22+135δβ2A3-1395A5δβ23+90A2δβ12σ2+2430A5δ2β2+1620δβ2γ2A5-1305A4δβ22+1620A4γ2δ+1215A4δ2σ22β22)Acos⁡(x)6720(1+σ1)4}t8+{-Acos⁡6(x)6720(1+σ1)4(72576A5δ2β2σ1+72576A5δ2β2)+[-Acos⁡5(x)6720(1+σ1)4(-99792A5γδ2σ1-99792A5γδ2)-Acos⁡2(x)6720(1+σ1)4(β221008A3δβ1β2+1344A2γ2β1-1344A2γβ2σ2σ1-1008A2δβ1σ2+1344A3γβ22+1344A2γβ2σ1+1344A3γβ22σ1+1344A2γβ2-1008A2δβ1σ1σ2+1344A2γ2β1σ1+12096A4δ2β1+1008A3δβ1σ1β2+12096A4δ2β1σ1-1344A2γβ2σ2+40320A4γβ2δσ1+40320A4γβ2δ+1008A2δβ1+1008A2δβ1σ1β22)-A6720(1+σ1)4(β22-448A2γβ2+448A2γβ2σ2σ1-448A2γβ2σ1-448A3γβ22σ1+448A2γβ2σ2+336A2δβ1σ2-1008A4δ2β1-336A3δβ1β2-336A2δβ1-336A3δβ1σ1β2+336A2δβ1σ1σ2-336A2δβ1σ1-4032A4γβ2δ-224A2γ2β1-448A3γβ22-4032A4γβ2δσ1-224A2γ2β1σ1-1008A4δ2β1σ1β22)-Acos⁡(x)6720(1+σ1)4(β222688A3δβ1β2+1680A3γβ22σ1-1792A3γ3+2688A3δβ1σ1β2+6720A3γδσ2σ1-6720A3γδσ1+6720A3γδσ2-31248A5γδ2-224A2γβ2σ1+224A2γβ2σ2σ1+1680A3γβ22+224A2γβ2σ2-1792A3γ3σ1-6720A3γδ-224A2γβ2-112Aγ+112Aγβ12+112Aγβ12σ1-112Aγσ22σ1+224Aγσ2σ1-112Aγσ22+224Aγσ2-112Aγσ1-6720A4γβ2δσ1-31248A5γδ2σ1-6720A4γβ2δβ22)-Acos⁡3(x)6720/(1+σ1)4(β22-3584A3γβ22σ1+9408A4γβ2δσ1+118944A5γδ2+9408A4γβ2δ+3584A3γ3σ1+9408A3γδ+9408A3γδσ1+118944A5γδ2σ1-3584A3γβ22-5376A3δβ1β2+3584A3γ3-9408A3γδσ2-5376A3δβ1σ1β2-9408A3γδσ2σ1β22)-Acos⁡4(x)6720(1+σ1)4(-15120A4δ2β1-47040A4γβ2δ-15120A4δ2β1σ1-47040A4γβ2δσ1β22)Acos⁡(x)6720(1+σ1)4]sin⁡(x)-Acos⁡7(x)6720(1+σ1)4(95256A6δ3σ1+95256A6δ3)-Acos⁡3(x)6720(1+σ1)4(β22-24192A4δ2σ2-24192A4δ2σ1σ2+24192δ2A4-1344A2γβ1β2σ1+122472A6δ3+1008A3β2δσ1-1344A2γβ1β2-18312A4δβ22σ1+1344A3γ2β2σ1-504A2δβ12+1344A2γ2σ1-1008A3δβ2σ2+24192A5δ2β2σ1+504A2δσ1+24192A4δ2σ1+54024A4γ2δσ1-1344A2γ2σ2+504A2δσ22-1008A2σ2δ+1008A2σ2δσ1+504A2δ-1008A3δβ2σ1σ2-1344A2γ2σ2σ1-122472A6δ3σ1+1008δβ2A3+1344A2γ2+504A2δσ1σ22-504A2δβ12σ1+24192A5δ2β2-18312A4δβ22+45024A4γ2δ+1344A3γ2β2β22)-Acos⁡5(x)6720(1+σ1)4(β22-195048A6δ3+15120A4δ2σ1σ2+15120A4δ2σ2-195048A6δ3σ1-15120A5δ2β2+13440A4δβ22-33600A4γ2δσ1-33600A4γ2δ-15120δ2A4-15120A4δ2σ1-15120A5δ2β2σ1+13440A4δβ22σ1β22)-A6720(1+σ1)4(β22-112Aγβ1σ1-1344δβ2A3-112A2γβ1β2-1008A3γβ1δσ1+112Aγβ1σ2σ1+1344A3δβ2σ1σ2-896A3γ2β2-112A2γβ1β2σ1-1344A3β2δσ1-1344A4δβ22σ1-1008A3γβ1δ-112Aγβ1+112Aγβ1σ2-896A3γ2β2σ1-4032A5δ2β2-1344A4δβ22+1344A3δβ2σ2-4032A5δ2β2σ1β22)-Acos⁡4(x)6720(1+σ1)4(β22-7168A3γ2β2-5376A4δβ22σ1-9408A3γβ1δ-9408A3γβ1δσ1-5376A3β2δσ1-7168A3γ2β2σ1-5376δβ2A3-124992A5δ2β2-5376A4δβ22+5376A3δβ2σ2+5376A3δβ2σ1σ2-124992A5δ2β2σ1β22)-Acos⁡(x)6720(1+σ1)4(β229072A4δ2σ2+9072A4δ2σ1σ2-9072δ2A4+896A2γβ1β2σ1-22680A6δ3-1008A3β2δσ1+896A2γβ1β2+4872A4δβ22σ1-1120A3γ2β2σ1+336A2δβ12-1120A2γ2σ1+1008A3δβ2σ2-9072A5δ2β2σ1-504A2δσ1-9072A4δ2σ1-13440A4γ2δσ1+1120A2γ2σ2-504A2δσ22+1008A2σ2δ+1008A2σ2δσ1-504A2δ+1008A3δβ2σ1σ2+1120A2γ2σ2σ1-22680A6δ3σ1-1008δβ2A3-1120A2γ2-504A2δσ1σ22+336A2δβ12σ1-9072A5δ2β2+4872A4δβ22-13440A4γ2δ-1120A3γ2β2β22)-Acos⁡2(x)6720(1+σ1)4(β22-224Aγβ1σ2σ1+224Aγβ1σ1+224A2γβ1β2+56448A5δ2β2+6720A4δβ22+9072A3γβ1δσ1+224Aγβ1+6720δβ2A3+9072A3γβ1δ+224A2γβ1β2σ1+7168A3γ2β2+56448A5δ2β2σ1+6720A3β2δσ1+7168A3γ2β2σ1-6720A3δβ2σ1σ2-224Aγβ1σ2+6720A4δβ22σ1-6720A3δβ2σ2β22)Acos⁡(x)6720(1+σ1)4}t6+{-Acos⁡3(x)6720(1+σ1)4(β22-1008A2β2β1ασ12-4032A2β22σ1-1008A2β2β1α-2016A2β2β1ασ1-2016A2β22-2016A2β22σ12β2σ12)-Acos⁡(x)6720(1+σ1)4(β221344A2β22σ12+672A2αβ2β1+1344A2β22+2688A2β22σ1+1344A2β2β1ασ1+672A2β2β1ασ12β2σ12)-A6720(1+σ1)4(β22-504δβ2A3-336A2β22σ1-168Aβ2σ12+336Aβ2σ2σ1+168Aβ2σ2-1008A3β2δσ1-168β2A-504A3β2δσ12-168A2β22-672A3β2γασ12-168A2β22σ12-336Aβ2σ1+168Aβ2σ2σ12-672A3β2γα-1344A3β2γασ1β2σ12)-Acos⁡2(x)6720(1+σ1)4(β225544A3β2δσ12+5376A3β2γασ12-672Aβ2σ2σ1-336Aβ2σ2+336β2A+672A2β22σ1+336A2β22+5376A3β2γα+672Aβ2σ1-336Aβ2σ2σ12+336Aβ2σ12+10752A3β2γασ1+336A2β22σ12+5544δβ2A3+11088A3β2δσ1β2σ12)+[-Acos⁡(x)6720(1+σ1)4(β22672β2Aβ1σ1+336β2Aβ1σ12+2688A3β22ασ12+2688A3β22α+336β2Aβ1+5376A3β22ασ1β2σ12)-Acos⁡2(x)6720(1+σ1)4(β222016A2γβ2+24192A4β2δασ1+2016A2γβ2σ12+2016A2β2σ1α+1008A3β22α+12096A4β2δασ12+1008A3β22ασ12+4032A2γβ2σ1-1008A2β2σ2ασ12+2016A3β22ασ1-2016A2β2σ2ασ1+12096A4β2δα+1008A2β2σ12α+1008A2β2α-1008A2β2σ2αβ2σ12)-A6720(1+σ1)4(β22-336A2γβ2-336A2β2α-2016A4β2δασ1-1008A4β2δασ12-1008A4β2δα-336A2β2σ12α-672A2γβ2σ1+336A2β2σ2α-336A3β22ασ12+672A2β2σ2ασ1+336A2β2σ2ασ12-336A3β22α-672A3β22ασ1-336A2γβ2σ12-672A2β2σ1αβ2σ12)-Acos⁡4(x)6720(1+σ1)4(-30240A4β2δασ1-15120A4β2δα-15120A4β2δασ12)-Acos⁡3(x)6720(1+σ1)4(-5376A3β22ασ12-5376A3β22α-10752A3β22ασ1)]sin⁡(x)-Acos⁡4(x)6720(1+σ1)4(β22-5376A3β2γα-12096A3β2δσ1-6048δβ2A3-6048A3β2δσ12-10752A3β2γασ1-5376A3β2γασ12β2σ12)Acos⁡(x)6720(1+σ1)4}t5+{-cos⁡2(x)6720(1+σ1)4(β225600Aβ2σ2σ1-26040A3β2δσ12-5600Aβ2σ1-2800A2β22-2800A2β22σ12-2800β2A-52080A3β2δσ1+2800Aβ2σ2σ12-26040δβ2A3-2240Aγβ1σ1-1120Aγβ1-5600A2β22σ1-2800Aβ2σ12-1120Aγβ1σ12+2800Aβ2σ2β2σ12)-Acos⁡(x)6720(1+σ1)4(β22280+560σ1-560σ2+22680δ2A4-560σ2σ12+560β2A+1120Aβ2σ1-560Aβ2σ2+560Aβ2σ12-1960A2β22σ12+8400A3β2δσ12+16800A3β2δσ1+5600A2γ2σ12+560σ22σ1+280σ22σ12-560Aβ2σ2σ12+280σ22+11200A2γ2σ1+16800A2δσ1+45360A4δ2σ1+22680A4δ2σ12-8400A2σ2δ-8400A2σ2δσ12-16800A2σ2δσ1+8400A2δσ12+280σ12+8400A2δ-1120σ2σ1+8400δβ2A3+5600A2γ2-1960A2β22-1120Aβ2σ2σ1-3920A2β22σ1β2σ12)-Acos⁡4(x)6720(1+σ1)4(23520A3β2δσ12+47040A3β2δσ1+23520δβ2A3)-Acos⁡3(x)6720(1+σ1)4(β2210080A2σ2δ-6720A2γ2-13440A2γ2σ1+10080A2σ2δσ12+3360A2β22-10080A2δσ12-60480δ2A4-10080A2δ-6720A2γ2σ12-120960A4δ2σ1-20160A3β2δσ1-10080δβ2A3-20160A2δσ1+20160A2σ2δσ1+6720A2β22σ1+3360A2β22σ12-60480A4δ2σ12-10080A3β2δσ12β2σ12)+[-Acos⁡(x)6720(1+σ1)4(β22-560β2Aβ1+23520A3γδσ12+23520A3γδ-1120β2Aβ1σ1+3360Aγσ12-3360Aγσ2σ12+6720Aγσ1-560β2Aβ1σ12+47040A3γδσ1+3360A2γβ2σ12-3360Aγσ2+3360A2γβ2+6720A2γβ2σ1-6720Aγσ2σ1+3360Aγβ2σ12)-A6720(1+σ1)4(β22560β1σ1-280σ2β1-280σ2β1σ12-560σ2β1σ1+280β1+1680A2δβ1σ1+280β2Aβ1σ12+560β2Aβ1σ1+2800A2γβ2+280β1σ12+2800A2γβ2σ12+5600A2γβ2σ1+840A2δβ1σ12+280β2Aβ1+840A2δβ1β2σ12)-Acos⁡3(x)6720(1+σ1)4(-33600A3γδ-67200A3γδσ1-33600A3γδσ12)-Acos⁡2(x)6720(1+σ1)4(β22-10080A2γβ2-20160A2γβ2σ1-5040A2δβ1σ1-2520A2δβ1σ12-10080A2γβ2σ12-2520A2δβ1β2σ12)Acos⁡(x)6720(1+σ1)4]sin⁡(x)-Acos⁡5(x)6720(1+σ1)4(37800δ2A4+37800A4δ2σ12+75600A4δ2σ1)-A6720(1+σ1)4(β224200A3β2δσ12+2800A2β22σ1-1400Aβ2σ2σ12+1400Aβ2σ12-2800Aβ2σ2σ1+1400β2A+1400A2β22σ12+560Aγβ1σ12+1400A2β22+8400A3β2δσ1+4200δβ2A3-1400Aβ2σ2+2800Aβ2σ1+1120Aγβ1σ1+560Aγβ1β2σ12)Acos⁡(x)6720(1+σ1)4}t4+{[-Acos⁡(x)6720(1+σ1)4(2240ΩAγ+4480ΩAγσ1+2240ΩAγσ12)-Acos⁡2(x)6720(1+σ1)4(-6720A2β2σ13α-20160A2β2σ12α-6720A2β2α-20160A2β2σ1α)-A6720(1+σ1)4(β226720A2β2σ1α+1120Ωβ1σ12+1120β1σ13+1120β1+3360β1σ12+2240Ωβ1σ1+3360β1σ1+1120Ωβ1+6720A2β2σ12α+2240A2β2σ13α+2240A2β2αβ2σ12)Acos⁡(x)6720(1+σ1)4]sin⁡(x)-Acos⁡2(x)6720(1+σ1)4(β22-4480ΩAβ2σ1-2240ΩAβ2σ12-2240β2A-6720Aβ2σ12-13440Aβ1ασ12-2240Aβ2σ13-4480Aβ1α-2240ΩAβ2-13440Aβ1ασ1-6720Aβ2σ1-4480Aβ1ασ13β2σ12)-A6720(1+σ1)4(β221120β2A+1120ΩAβ2σ12+1120Aβ2σ13+2240Aβ1α+2240Aβ1ασ13+1120ΩAβ2+3360Aβ2σ12+2240ΩAβ2σ1+6720Aβ1ασ1+3360Aβ2σ1+6720Aβ1ασ12β2σ12)-Acos⁡(x)6720(1+σ1)4(β223360ΩA2δ+1120ΩAβ2-2240Ωσ2σ1-1120Ωσ2-1120Ωσ2σ12+1120Ω+2240Ωσ1+2240ΩAβ2σ1+1120Ωσ12+6720ΩA2δσ1+3360ΩA2δσ12+1120ΩAβ2σ12β2σ12)-Acos⁡3(x)6720(1+σ1)4(-6720ΩA2δσ1-3360ΩA2δσ12-3360ΩA2δ)Acos⁡(x)6720(1+σ1)4}t3+{-Acos⁡3(x)6720(1+σ1)4(30240A2δσ12+10080A2δσ13+10080A2δ+30240A2δσ1)-A6720(1+σ1)4(-10080Aβ2σ12-3360β2A-3360Aβ2σ13-10080Aβ2σ1)+[-A6720(1+σ1)4(-10080β1σ12-3360β1-3360β1σ13-10080β1σ1)-Acos⁡(x)6720(1+σ1)4(-20160Aγσ1-6720Aγσ13-6720Aγ-20160Aγσ12)]sin⁡(x)-Acos⁡2(x)6720(1+σ1)4(20160Aβ2σ1+20160Aβ2σ12+6720Aβ2σ13+6720β2A)-A/cos⁡(x)6720(1+σ1)4(β22-30240A2δσ12-10080Aβ2σ1+3360σ2σ13-10080A2δ+10080σ2σ1-3360Aβ2σ13-3360-30240A2δσ1-3360β2A+10080σ2σ12+3360σ2-10080σ12-10080A2δσ13-10080Aβ2σ12-3360σ13-10080σ1β2σ12)Acos⁡(x)6720(1+σ1)4}t2-Acos⁡(x)6720(1+σ1)4(6720+6720σ14+40320σ12+26880σ13+26880σ1)-A6720(1+σ1)4(-6720-26880σ1-26880σ13-6720σ14-40320σ12),(5.5)θ2={-Acos⁡(x)524(1+σ1)2(-540A4δα2-540A4δα2σ1+216A4bδβ2)-Acos⁡(x)324(1+σ1)2(β22-24A3bβ22+36A3β2α2+24A2bσ2β2+36A3β2α2σ1+36A2α2σ1+36A2αγσ1+702A4δα2σ1-36A2σ2α2σ1-36A2σ2α2+36A2α2-24A2bβ1γ-288A4bδβ2-24A2bβ2+36A2αγ+702A4δα2β2σ12)+[-Acos⁡2(x)24(1+σ1)2(β22-24A2bγσ2+36A2β1α2σ1-24A2bβ1β2+36A2β1α2+36A2β2α+24A3bγβ2+24A2bγ+180A4bγδ+36A2β2σ1αβ2σ12)+9A5bγδcos⁡4(x)(1+σ1)2-Acos⁡(x)24(1+σ1)2(β22-6αAσ1+6bAβ1+18A3bβ1δ-72αA3δ-96A3γα2σ1+48A3bγβ2-6A2β2σ1α-6A2β2α-96A3γα2+6αAσ2-6αA-6Abβ1σ2-72αA3δσ1+6αAσ2σ1+6A2bβ1β2β2σ12)-Acos⁡3(x)24(1+σ1)2(192A3γα2σ1+192A3γα2+108αA3δ-96A3bγβ2+108αA3δσ1-54A3bβ1δ)-A24(1+σ1)2(β22-12A2bγ-12A2β2α-12A3bγβ2-36A4bγδ-6A2β1α2-12A2β2σ1α+12A2bγσ2-6A2β1α2σ1β2σ12)Acos⁡(x)6720(1+σ1)4]sin⁡(x)-Acos⁡2(x)24(1+σ1)2(β223Ab+72A4bδβ2+48A3bγ2+3Abσ22+189A5bδ2+6Aβ1ασ1-6A2bσ2β2+6Aβ1α-3Abβ12+192A3β2α2+192A3β2α2σ1+72A3bδ+6A2bβ2-45A3bβ22-72A3bδσ2-6Abσ2β2σ12)-Acos⁡(x)24(1+σ1)2(β2272A4bδβ2-198A4δα2σ1-30A3β2α2-30A2α2σ1+24A3bβ22-30A3β2α2σ1+30A2σ2α2-24A2bσ2β2+12A2bβ1γ+30A2σ2α2σ1-198A4δα2-30A2αγσ1+24A2bβ2-30A2αγ-30A2α2β2σ12)-Acos⁡4(x)24(1+σ1)2(β22-405A5bδ2-192A3β2α2+54A3bδσ2-54A4bδβ2-48A3bγ2+48A3bβ22-54A3bδ-192A3β2α2σ1β2σ12)-81A6bδ2cos⁡6(x)8(1+σ1)2-A24(1+σ1)2(β22-12A3bγ2-18A4bδβ2-3Ab-3Abσ22-3A3bβ22+6A2bσ2β2-3Aβ1α-27A5bδ2-3Aβ1ασ1-24A3β2α2σ1-18A3bδ-6A2bβ2+18A3bδσ2-24A3β2α2+6Abσ2β2σ12)Acos⁡(x)6720(1+σ1)4}t4+{-A(-48A2αγ-48A2αγσ1)cos⁡3(x)24(1+σ1)2-A(40A2αγ+40A2αγσ1)cos⁡(x)24(1+σ1)2-A(4Aβ1α+4Aβ1ασ1)24(1+σ1)2-A(-8Aβ1α-8Aβ1ασ1)cos⁡2(x)24(1+σ1)2+[-A(-48A2β2σ1α-48A2β2α)24(1+σ1)2cos⁡2(x)-A(-144αA3δσ1-144αA3δ)cos⁡3(x)24/(1+σ1)2-Acos⁡(x)24(1+σ1)2(β228αA+96αA3δσ1-8αAσ2σ1-8αAσ2+8A2β2α+96αA3δ+8A2β2σ1α+8αAσ1β2σ12)-A24(1+σ1)2(16A2β2α+16A2β2σ1α)]sin⁡(x)}t3+{-Acos⁡(x)24(1+σ1)2(β2212-48A2bβ2+120A2α2σ12-48A2σ1bβ2+24σ1+12σ12-12σ1aβ1+240A2α2σ1-12aβ1+120A2α2β2σ12)-Acos⁡3(x)24(1+σ1)2(48A2bβ2-144A2α2σ12-288A2α2σ1-144A2α2+48A2σ1bβ2)-A24(1+σ1)2(β2224Aaγ+36A3bδ+12A2bβ2-12Abσ2+12Ab+36A3σ1bδ+24Aσ1aγ+12Aσ1b-12Aσ1bσ2+12A2σ1bβ2β2σ12)+[-Acos⁡2(x)24(1+σ1)2(-48A2bγ-108aδA2-108σ1aδA2-48A2σ1bγ)-A24(1+σ1)2(β2212σ1aβ2A+36σ1aδA2-12aσ2+12aβ2A-12σ1aσ2+12σ1a+24A2bγ+36aδA2+12a+24A2σ1bγβ2σ12)-Acos⁡(x)24(1+σ1)2(β22240αAσ1+120αA-48aβ2A-12bAβ1-48σ1aβ2A+120αAσ12-12σ1bAβ1β2σ12)Acos⁡(x)6720(1+σ1)4]sin⁡(x)-Acos⁡2(x)24(1+σ1)2(β22-12Ab+12Abσ2-144A3σ1bδ-144A3bδ-12A2bβ2-48Aaγ-12A2σ1bβ2-48Aσ1aγ-12Aσ1b+12Aσ1bσ2β2σ12)-A(108A3bδ+108A3σ1bδ)cos⁡4(x)24(1+σ1)2}t2×{A(-48αA-96αAσ1-48αAσ12)cos⁡(x)sin⁡(x)24(1+σ1)2+-A(-24-48σ1-24σ12)cos⁡(x)24(1+σ1)2}t-A(24+48σ1+24σ12)cos⁡(x)24(1+σ1)2-A(-24-48σ1-24σ12)24(1+σ1)2.
Now we make calculations for the results obtained by the variational iteration method using the Maple software package with the following arbitrary constants:(5.6)a=0.5,A=0.001,b=0.5,α=1,β1=β2=0.05,γ=1,δ=0.8.


### 5.1. Special Case

(1)
If we take into our consideration the first iteration (i.e.,u=u1 and θ=θ1). See Figures 15, 16, 17, and 18.(2)
If the magnetic field and rotation are neglected, the components of the displacementu2 and temperature θ2 take the following forms. See Figures 19 and 20.


## 5.1. Special Case

(1)
If we take into our consideration the first iteration (i.e.,u=u1 and θ=θ1). See Figures 15, 16, 17, and 18.(2)
If the magnetic field and rotation are neglected, the components of the displacementu2 and temperature θ2 take the following forms. See Figures 19 and 20.


## 6. Discussion

With the view of illustrating the theoretical results obtained in the preceding sections, a numerical result is calculated for the homotopy perturbation method and variational iteration method.Figures (1–10) illustrate the influences of time t, rotation Ω, and sensitive pats of the magnetic field σ1 and σ2 for the iterations (u=V0+V1+V2andθ=Θ0+Θ1+Θ2) and (u=V0+V1andθ=Θ0+Θ1), and if the rotation and magnetic field neglected, respectively, respect to the coordinate x for the homotopy perturbation method. Figures (11–20) illustrate the influences of time t, rotation Ω, and sensitive pats of the magnetic field σ1 and σ2 for the iterations (u=V2, θ=θ2 and u=V1 and θ=θ1), and if the rotation and magnetic field have been neglected, respectively, respect to the coordinate x for the variational iteration method.Variations of the displacementu2 and temperature θ2 for various values of the axis x and time t when Ω=0.1, σ1=0.2, σ2=0.1.
(a)(b)Variations of the displacementu2 and temperature θ2 for various values of the axis x and rotation Ω when t=0.1, σ1=0.2, σ2=0.1.
(a)(b)Variations of the displacementu2 and temperature θ2 for various values of the axis x and magnetic field σ1 when t=0.1, Ω=0.1, σ2=0.1.
(a)(b)Variations of the displacementu2 and temperature θ2 for various values of the axis x and magnetic field σ2 when t=0.1, Ω=0.1, σ1=0.1.
(a)(b)Variations of the displacementu1 and temperature θ1 for various values of the axis x and time t when Ω=0.1, σ1=0.2, σ2=0.1.
(a)(b)Variations of the displacementu1 and temperature θ1 for various values of the axis x and rotation Ω when t=0.1, σ1=0.2, σ2=0.1.
(a)(b)Variations of the displacementu1 and temperature θ1 for various values of the axis x and magnetic field σ1 when t=0.1, Ω=0.1, σ2=0.1.
(a)(b)Variations of the displacementu1 and temperature θ1 for various values of the axis x and magnetic field σ2 when t=0.1, Ω=0.1, σ1=0.1.
(a)(b)Variations of the displacementu2 and temperature θ2 for various values of the axis x and time t (u=u2 and θ=θ2) when Ω=σ1=σ2=0.
(a)(b)Variations of the displacementu1 and temperature θ1 for various values of the axis x and time t (u=u1 and θ=θ1) when Ω=σ1=σ2=0.
(a)(b)From Figures1 and 11, it is concluded that the displacement u and temperature θ start from their maximum values, decrease and increase periodically with an increasing of the coordinate x, also, it is obvious that their values take the minimum values and increases with the increasing values of the time t. From Figures 2, 3, 4, 12, 13, 14, it is seen that the components of the displacement u and temperature θ begin from the minimum values near zero increase and then decrease periodically with the coordinate x, it is clear also that there are a sligh increasing with an increasing of the sensitive parts of the magnetic field, also, one can see that u and θ decrease with an increasing of the rotation Ω.Figures5–8 and 15–18 display the first iteration with respect to the homotopy perturbation method and variational iteration method on the influences of the parameters time t, rotation Ω, and sensitive pats of the magnetic field σ1 and σ2 to obtain the displacement and the temperature components on the medium due to the harmonic wave propagation. It is shown that the increasing of the coordinate x sensitive an increasing and dereasing on them periodically due to appearance of the pairs (cos, sin) in the initial condition and the approximate solutions; it is also clear that the components begin from their minimum values and increase absolutely with the variation of the time t. With the variations of the rotation and magnetic field tends to slightly affect on the displacment and the temperature.It seems too that there are a clear differs between the results obtained by the HPM and the corresponding results obtained by VIM resultant to the appearance of the high order of time in VIM tends to the high values of the approximate solution comparing with the results obtained by HPM. Because of the results obtained, we concluded that the homotopy perturbation method is more effective and powerful than the variational iteration method.On the other hand, from Figures9, 10, 19, and 20, it is obvious that if the rotaion and magnetic field are neglected, the approximate solutions by HPM and VIM in first iteration are the same in both methods and agree with the results obtained by Sweilam and Khader [1].



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*Source: 827901-2012-03-07.xml*