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

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-spacemodel subjected initially to a prescribed harmonic displacement and the temperature of themedium. 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.


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 everincreasing 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.

Basic Idea of He's Homotopy Perturbation Method
We illustrate the following nonlinear differential equation 8, 22 : with the boundary conditions: 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: By the homotopy technique, we construct a homotopy V r, p : where p ∈ 0, 1 is an embedding parameter and u 0 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
The changing process of p from zero to unity is just that of V r, p from u 0 r to u r .In topology, this is called deformation, and L V − L u 0 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: On setting p 1 results in the approximate solution of 2.3 , we have 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 V must be small because the parameter may be relatively large, that is, p → 1.

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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: where γ, β 1 , β 2 , a, b, α are arbitrary constants, σ 1 , σ 2 are the sensitive parts of the magnetic field, and Ω is the rotation parameter, with the initial conditions where A is an arbitrary constant and the boundary conditions u 0, t θ 0, t 0, u t 0, t θ t 0, t 0.

3.3
To investigate the traveling wave solution of 3.1 , we first construct a homotopy perturbation method as follows: where the initial approximations take the following form:

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: where V j and Θ j , 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 In order to obtain the unknowns of V j and Θ j , j 1, 2, 3, . . ., we construct and solve the following system considering the initial conditions 3.2 :

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Consequently, we deduce after some calculations the following results: where

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:

3.11
The results obtained in 3.9 are displayed graphically in Figures 1-4.

Basic Idea of Variational Iteration Method
Consider the following nonhomogeneous nonlinear system of partial differential equations: where L 1 , L 2 are linear differential operators with respect to time, N 1 , N 2 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: where λ 1 and λ 2 are 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, u n and θ n are restricted variations.We can assume that the above correctional functionals are stationary i.e., δu n 1 0 and δθ n 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.

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: where u n and θ n are considered as a restricted variation, that is, δ u n 1 0 and δ θ n 1 0. Consequently, the general Lagrange multipliers λ 1 and λ 2 take the following form: By the substitution of the identified Lagrange multipliers 5.2 into 5.1 , we have the following iteration relations:

5.3
With help of Maple or Mathematica, we get the following results: Mathematical Problems in Engineering

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 σ 1 and σ 2 for the iterations u 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 σ 1 and σ 2 for the iterations u V 2 , θ θ 2 and u V 1 and θ θ 1 , and if the rotation and magnetic field have been neglected, respectively, respect to the coordinate x for the variational iteration method.
From Figures 1 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.
Figures 5-8 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 σ 1 and σ 2 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.

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On the other hand, from Figures 9,10,19, 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 .

Figure 1 :Figure 2 :
Figure 1: Variations of the displacement u and temperature θ for various values of the axis x and time t when Ω 0.1, σ 1 0.2, σ 2 0.1.

Figure 3 :
Figure 3: Variations of the displacement u and temperature θ for various values of the axis x and magnetic field σ 1 when t 0.1, Ω 0.1, σ 2 0.1.

Figure 4 :Figure 5 :Figure 6 :
Figure 4: Variations of the displacement u and temperature θ for various values of the axis x and magnetic field σ 2 when t 0.1, Ω 0.1, σ 1 0.1.

Figure 7 :Figure 8 :
Figure 7: Variations of the displacement u and temperature θ for various values of the axis x and magnetic field σ 1 when t 0.1, Ω 0.1, σ 2 0.1.

Figure 9 :
Figure 9: Variations of the displacement u and temperature θ for various values of the axis x and time tu V 0 V 1 V 2 and θ Θ 0 Θ 1 Θ 2 when Ω σ 1 σ 2 0.

Figure 10 :
Figure 10: Variations of the displacement u and temperature Θ for various values of the axis x and time t u V 0 V 1 and θ Θ 0 Θ 1 when Ω σ 1 σ 2 0.

Figure 11 :Figure 12 :
Figure 11: Variations of the displacement u 2 and temperature θ 2 for various values of the axis x and time t when Ω 0.1, σ 1 0.2, σ 2 0.1.

Figure 13 :Figure 14 :
Figure 13: Variations of the displacement u 2 and temperature θ 2 for various values of the axis x and magnetic field σ 1 when t 0.1, Ω 0.1, σ 2 0.1.

Figure 15 :Figure 16 :
Figure 15: Variations of the displacement u 1 and temperature θ 1 for various values of the axis x and time t when Ω 0.1, σ 1 0.2, σ 2 0.1.

Figure 17 :
Figure 17: Variations of the displacement u 1 and temperature θ 1 for various values of the axis x and magnetic field σ 1 when t 0.1, Ω 0.1, σ 2 0.1.

Figure 18 :Figure 19 :
Figure 18: Variations of the displacement u 1 and temperature θ 1 for various values of the axis x and magnetic field σ 2 when t 0.1, Ω 0.1, σ 1 0.1.

Figure 20 :
Figure 20: Variations of the displacement u 1 and temperature θ 1 for various values of the axis x and time t u u 1 and θ θ 1 when Ω σ 1 σ 2 0.