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.
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 , 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  investigated variational iteration method for one dimensional nonlinear thermoelasticity. Applying He’s variational iteration method for solving differential-difference equation is discussed by Yildirim . Noor and Mohyud-Din , 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.  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.  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]:
with the boundary conditions:
where is a general differential operator, is a boundary operator, is an analytic function, and is the boundary of the domain . Generally speaking, the operator can be divided into two parts which are and , where is linear operator but is nonlinear operator. Equation (2.1) can therefore be rewritten as follows:
By the homotopy technique, we construct a homotopy : which satisfies
where is an embedding parameter and is an initial approximation of (2.1) which satisfies the boundary conditions (2.2). Obviously, from (2.4) and (2.5) we have
The changing process of from zero to unity is just that of from to . In topology, this is called deformation, and and are called homotopy. According to the homotopy perturbation method, we can first use the embedding parameter “” as a small parameter and assume that the solution of (2.4) and (2.5) can be written as a power series in “” as follows:
On setting 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 .(1)The second derivative of with respect to must be small because the parameter may be relatively large, that is, .(2)The norm of 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:
where are arbitrary constants, are the sensitive parts of the magnetic field, and is the rotation parameter, with the initial conditions
where is an arbitrary constant and the boundary conditions
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:
According to the homotopy perturbation method, we can first use the embedding parameter “” as a small parameter and assume that the solution of (3.4) can be written as a power series in “” as the following:
where and , are functions to be determined.
Substituting from (3.6) into (3.4) and arranging the coefficients of “” powers, we have
In order to obtain the unknowns of and , , we construct and solve the following system considering the initial conditions (3.2):
Consequently, we deduce after some calculations the following results:
Now we make calculations for the results obtained by the homotopy perturbation method using the Maple software package with the following arbitrary constants:
The results obtained in (3.9) are displayed graphically in Figures 1–4.
Figure 1: Variations of the displacement and temperature for various values of the axis and time when, , .
Figure 2: Variations of the displacement and temperature for various values of the axis and rotation when , , .
Figure 3: Variations of the displacement and temperature for various values of the axis and magnetic field when, , .
Figure 4: Variations of the displacement and temperature for various values of the axis and magnetic field when , , .
3.1. Special Cases
(1)If we take into our consideration the first iteration (i.e., and ). See Figures 5, 6, 7, and 8.(2)If the magnetic field and rotation are neglected, the components of the displacement and temperature take the following forms. See Figures 9 and 10.
Figure 5: Variations of the displacement and temperature for various values of the axis and time when , , .
Figure 6: Variations of the displacement and temperature for various values of the axis and rotation when , , .
Figure 7: Variations of the displacement and temperature for various values of the axis and magnetic field when , , .
Figure 8: Variations of the displacement and temperature for various values of the axis and magnetic field when , , .
Figure 9: Variations of the displacement and temperature for various values of the axis and time ( and ) when .
Figure 10: Variations of the displacement and temperature for various values of the axis and time ( and ) when .
4. Basic Idea of Variational Iteration Method
Consider the following nonhomogeneous nonlinear system of partial differential equations:
where , are linear differential operators with respect to time, , are nonlinear operators, and , are given functions.
According to the variational iteration method, we can construct correct functionals as follows:
where and 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 denotes the th order approximation, and are restricted variations. We can assume that the above correctional functionals are stationary (i.e., and ), 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:
where and are considered as a restricted variation, that is, and . Consequently, the general Lagrange multipliers and take the following form:
By the substitution of the identified Lagrange multipliers (5.2) into (5.1), we have the following iteration relations:
With help of Maple or Mathematica, we get the following results: