Analytic Solution for a Nonlinear Problem of Magneto-Thermoelasticity

REPORTS ON MATHEMATICAL PHYSICS

Vol. 71 (2013)

No. 3

ANALYTIC SOLUTION FOR A NONLINEAR PROBLEM OF MAGNETO-THERMOELASTICITY A. JAFARIANa , P. G HADERIa , A LIREZA K. G OLMANKHANEHb and D. BALEANUc,d,e a Department

of Mathematics, Urmia Branch, Islamic Azad University, Urmia, Iran (e-mail: [email protected])

b Department

of Physics, Urmia Branch, Islamic Azad University, P.O. BOX 969, Oromiyeh, Iran (e-mail: [email protected])

cC ¸ ankaya

University, Faculty of Art and Sciences, Department of Mathematics and Computer Sciences, Balgat 0630, Ankara, Turkey d Department

of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box: 80204, Jeddah 21589, Saudi Arabia e Institute of Space Sciences, P.O. BOX, MG-23, R 76900, Magurele-Bucharest, Romania (e-mail: [email protected])

(Received October 29, 2012 – Revised March 26, 2013) In this paper, we present a comparative study of the homotopy analysis method (HAM), the variational iteration method (VIM) and the iterative method (He’s polynomials). The approximate solution of the coupled harmonic waves nonlinear magneto-thermoelasticity equations under influence of rotation is obtained. In order to control and adjust the convergence region and the rate of solution series, we show that it is possible to choose a valid auxiliary parameter h of HAM. Using the boundary and the initial conditions we select a suitable initial approximation. The results show that these methods are very efficient, convenient and applicable to a large class of problems. Keywords: homotopy analysis method, variational iteration method, approximate solution, h-curve, nonlinear coupled magneto-thermoelasticity.

1.

Introduction

Mathematical modeling of many physical systems leads to nonlinear ordinary and partial differential equations. The multi-dimensional coupled systems of parabolic and hyperbolic equations often arise in the study of circled fuel reactor, hightemperature hydrodynamics, thermoelasticity and magneto-thermoelastic problems [1–4]. As a result, the analysis of this type of models was done using a wide [399]

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range of analytical and numerical methods. Researchers use analytical techniques to study nonlinear problems. Thus, finding any analytic solution of nonlinear differential equations plays a prominent role in mathematics. A variety of effective analytical and semi-analytical methods were developed and used for solving nonlinear differential equations, i.e. the differential transform method [5–15] and the Adomian decomposition method (ADM) [16–18]. In 1992 Liao [18] presented a homotopy analysis method, allowing for more than uniformly valid analytical solution of nonlinear equation with no possible small parameters. Liao successfully applied the HAM to solve many types of nonlinear problems [18]. The HAM includes a certain auxiliary parameter h which provides a simple way to adjust and control the convergent region and the rate of convergence of the series solution. Moreover, it is possible to find the convergence series solution by choosing the valid region corresponding to the plot of the so-called h-curve. We notice that VIM was first proposed by He in [19, 20]. The idea of VIM is to construct an iteration method based on a correction functional that includes a generalized Lagrange multiplier. Also, an iterative method (He’s polynomials) is applied for solving linear and nonlinear functional equations [21]. As a result of these works, we conclude that HAM, VIM are powerful methods to solve nonlinear problems [23–26] and present certain advantages over some other numerical methods. This paper is arranged as follow. The model of the problem is presented in Section 2. In Section 3, the basic ideas of the present approaches are described. In Section 4, by choosing special forms of initial conditions, the proposed methods are applied to study nonlinear equations of magneto-thermoelasticity in the presence of rotation. In Section 5, numerical results are given and the accuracy of the proposed methods is discussed by comparison between HAM, VIM and He’s polynomials. Finally, conclusions are reported in Section 6. 2.

The model

Multi-dimensional coupled systems of parabolic and hyperbolic equations appear in the study of circled fuel reactor, high-temperature hydrodynamics, thermoelasticity and magneto-thermoelasticity. In this paper we solve nonlinear equations of magnetothermoelasticity with rotation, namely [26] ⎧ ⎨ (1 + σ )u + u − u (1 − σ + 2γ u + 3δu2 ) − β θ − β (θ u ) = 0, 1 tt t xx 2 x 1 x 2 x x x (1) ⎩ (θ − aux − 1 bu2 )t − [(1 + αux )θx ]x = 0, 2

x

under the initial conditions u(x, 0) = θ(x, 0) = A(1 − cos(x)),

ut (x, 0) = θt (x, 0) = 0,

(2)

and the boundary conditions u(0, t) = θ(0, t) = 0,

ut (0, t) = θt (0, t) = 0,

(3)

401

ANALYTIC SOLUTION FOR A NONLINEAR PROBLEM. . .

where γ , β1 , β2 , a, b and α are arbitrary constants, σ1 and σ2 are the sensitive parts of magnetic field,  is the rotation parameter and A is an arbitrary constant. The symbols u = u(x, t) and θ = θ(x, t) in (1) denote the displacement and the temperature, respectively. Here, x denotes the spatial coordinate and t denotes the time coordinate. 3.

The methods

3.1.

The homotopy analysis

Let us consider the following differential equation N [u(x, t)] = 0,

(4)

where N is a nonlinear operator, t is independent variable, and u(x, t) is an unknown function. Using the homotopy analysis, an auxiliary parameter h can adjust and control the convergence region of homotopy-series solutions; this parameter introduces one more degree of freedom [18]. The generalized deformation equation was introduced as follows, [1 − α(q)]L[φ(x, t; q) − u0 (x, t)] = hβ(q)H (x, t)N[φ(x, t; q)],

q ∈ [0, 1], (5)

where α(q) and β(q) are the so-called deformation functions satisfying α(0) = β(0) = 0, where α(q) =

∞ 

αm q m ,

α(1) = β(1) = 1, β(q) =

m=1

∞ 

αm q m .

m=1

Using the zeroth-order deformation equation, namely [18] (1 − q)L[φ(x, t; q) − u0 (x, t)] = qhH (x, t)N[φ(x, t; q)],

(6)

where q ∈ [0, 1] is an embedding parameter, h  = 0 is a nonzero auxiliary parameter, H (x, t)  = 0 is a nonzero auxiliary function, L is an auxiliary linear operator, u0 (x, t) is initial guess of u(x, t), and φ(x, t; q) is an unknown function. We notice that when q = 0 and q = 1, it implies φ(x, t; 0) = u0 (x, t),

φ(x, t; 1) = u(x, t).

(7)

Therefore, if q increases from 0 to 1, the solution φ(x, t; q) varies from u0 (x, t) to the solution u(x, t). Using the Taylor series with respect to q, we obtain φ(x, t; q) = u0 (x, t) +

∞ 

um (x, t) q m ,

(8)

m=1

where

 1 ∂ m φ(x, t; q)  um (x, t) =  . m! ∂q m q=0

(9)

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If the auxiliary linear operator, the initial guess, the auxiliary parameter h, and the auxiliary function are properly chosen, the series (8) converges at q = 1, so we have ∞  u(x, t) = u0 (x, t) + um (x, t), (10) m=1

which is one of the solutions for the nonlinear equation [18]. For the particular case h = −1 and H (x, t) = 1, Eq. (6) becomes (1 − q)L[φ(x, t; q) − u0 (x, t)] + qN [φ(x, t; q)] = 0,

(11)

which is called the classical homotopy perturbation method. According to the definition (9), the governing equation can be deduced from the zeroth-order deformation Eq. (6). Let us define a vector un = {u0 (x, t), u1 (x, t), . . . , un (x, t)}.

(12)

Differentiating Eq. (6) m times with respect to q, choosing q = 0 and dividing it by m!, we arrive at an m-th order deformation equation, namely um−1 ), L[um (x, t) − χm um−1 (x, t)] = hH (x, t)Rm ( such that um−1 ) = Rm (

∂ m−1 N [φ(x, t; q)] 1 |q=0 , (m − 1)! ∂q m−1 

and χm =

0, 1,

m ≤ 1, m > 1.

(13) (14)

(15)

So, um (x, t) for m ≥ 1 can be obtained by using Eq. (13), the boundary condition, and MATLAB computer software. 3.2.

Variational iteration

In this section we study the variational iteration method (VIM). We consider a general differential equation Lu + Nu = f (x), where L is a linear operator, N is a nonlinear operator and f (x) is the forcing term. Using the variational iteration method [28-31], we construct a correct functional as below  x λ(Lun (τ ) + N u˜ n (τ ) − f (τ )) dτ, (16) un+1 (x) = un (x) + 0

where λ is a Lagrange multiplier [28-31], which is identified optimally via the variational iteration method. The subscripts n denote the n-th approximation, u˜ n is considered as a restricted variation, i.e. δ u˜ n = 0; Eq. (16) is called a correct functional. The principles of the variational iteration method and its applicability for various kinds of differential equations are given in [28–34]. In the VIM method, at first we have to determine the Lagrange multiplier λ optimally. The successive

ANALYTIC SOLUTION FOR A NONLINEAR PROBLEM. . .

403

approximation un+1 , n ≥ 0 of the solution u will be readily obtained using the determined Lagrange multiplier and any selective function u0 , consequently, the solution is given by u = lim un . n→∞

3.3.

The iterative method (He’s polynomials)

Consider a general functional equation as [21] y = N(y) + f,

(17)

where N denotes a nonlinear operator from a Banach space B → B and f represents a known function. We are looking for a solution y of Eq. (17) having the following series form ∞  y= yi (t). (18) i=1

The nonlinear operator N is decomposed as ∞

 

 

i n i−1 N ) + = N(y . N − N y y y 0 i=1 i j =0 j j =0 j

(19)

i=1

From Eqs. (18), (19) and (17) we conclude that n ∞ 

  

i i−1 yi = f + N(y0 ) + . N − N y y j j j =0 j =0 i=1

(20)

i=1

So, the recurrence relation has the form ⎧ ⎪ ⎪ ⎨ y0 = f, y1 = N (y0 ), ⎪ ⎪ ⎩y m+1 = N(y0 + · · · + ym ) − N(y0 + · · · + ym−1 ),

(21) m = 1, 2, . . . .

Thus, we have (y0 + · · · + ym+1 ) = N(y0 + · · · + ym ), and

∞  i=1

yi = f + N

∞ 

 yj ,

m = 1, 2, . . . ,

(22) (23)

j =0

respectively. The k-term approximate solution of Eqs. (17) and (18) is given by

y = ki=0 yi . 4.

The main results In this section, we can start with the given initial approximation and using iteration formulae from the previous section, we can obtain approximate solutions.

404 4.1.

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The solution with HAM

To solve the system (1) within HAM, according to the initial conditions denoted in Eqs. (2), it is natural to choose u0 (x, t) = A(1 − cos(x)),

θ0 (x, t) = A(1 − cos(x)).

(24)

The expressions of the linear operators are ∂ 2 φ1 ∂φ2 , (25) , L2 [φ2 (x, t; q)] = ∂t 2 ∂t such that L1 [c1 + tc2 ] = 0, L2 [c3 ] = 0. Here c1 , c2 and c3 are constant coefficients. The nonlinear operators N1 and N2 are defined as L1 [φ1 (x, t; q)] =

N1 [φ1 , φ2 ] =

∂ 2 φ1  ∂φ1 σ2 ∂ 2 φ1 2γ ∂ 2 φ1 ∂φ1 1 ∂ 2 φ1 + + − − ∂t 2 1 + σ1 ∂t 1 + σ1 ∂x 2 1 + σ1 ∂x 2 1 + σ1 ∂x 2 ∂x   3δ ∂ 2 φ1 ∂φ1 2 β2 ∂φ2 ∂φ1 β2 ∂ 2 φ1 β1 ∂φ2 − − − − φ2 2 , 2 1 + σ1 ∂x ∂x 1 + σ1 ∂x 1 + σ1 ∂x ∂x 1 + σ1 ∂x

∂ 2 φ1 ∂ 2 φ1 ∂φ1 ∂ 2 φ2 ∂φ2 ∂φ1 ∂ 2 φ2 ∂ 2 φ1 ∂φ2 −a −b − . − α − α ∂t ∂t∂x ∂t∂x ∂x ∂x 2 ∂x ∂x 2 ∂x 2 ∂x Using the above definitions, together with the assumption H1 (x, t) = H2 (x, t) = 1, the zeroth-order deformation equations become N2 [φ1 , φ2 ] =

(1 − q)L1 [φ1 (x, t; q) − u0 (x, t)] = qhN1 [φ1 , φ2 ], (1 − q)L2 [φ2 (x, t; q) − θ0 (x, t)] = qhN2 [φ1 , φ2 ]. Obviously, when q = 0 and q = 1, φ1 (x, t; 0) = u0 (x, t), φ2 (x, t; 0) = θ0 (x, t),

φ1 (x, t; 1) = u(x, t), φ2 (x, t; 1) = θ (x, t).

Thus, we obtain the m-th order deformation equations, namely L1 [um (x, t) − χm um−1 (x, t)] = hR1m ( um−1 , θm−1 ), L2 [θm (x, t) − χm θm−1 (x, t)] = hR2m ( um−1 , θm−1 ),

(26) (27)

where ∂ 2 um−1 1 ∂ 2 um−1  ∂um−1 σ2 ∂ 2 um−1 − + + ∂t 2 1 + σ1 ∂t 1 + σ1 ∂x 2 1 + σ1 ∂x 2  m−1  m−1 n  ∂ 2 uj ∂um−1−j   ∂ 2 um−1−n   ∂uj ∂un−j 3δ 2γ − − 1 + σ1 j =0 ∂x 2 ∂x 1 + σ1 n=0 ∂x 2 ∂x ∂x j =0   m−1  m−1  ∂θj ∂um−1−j  ∂ 2 um−1−j  β2 β2 β1 ∂θm−1 − − θj , − 1 + σ1 ∂x 1 + σ1 j =0 ∂x ∂x 1 + σ1 j =0 ∂x 2

R1m ( um−1 , θm−1 ) =

ANALYTIC SOLUTION FOR A NONLINEAR PROBLEM. . .

R2m ( um−1 , θm−1 ) =

405

 m−1  ∂ 2 uj ∂um−1−j  ∂ 2 um−1 ∂θm−1 −a −b ∂t ∂t∂x ∂t∂x ∂x n=0  m−1   m−1 2  ∂uj ∂θm−1−j  ∂ 2 uj ∂θm−1−j  ∂ 2 θm−1 − −α . −α ∂x 2 ∂x ∂x 2 ∂x 2 ∂x j =0 j =0

Now, the solution of the m-th order deformation Eq. (26) and Eq. (27) for m ≥ 1 becomes u0 (x, t) = −A(cos(x) − 1), θ0 (x, t) = −A(cos(x) − 1), u1 (x, t) = −(h(4At 2 cos(x) − 3A3 δt 2 cos(3x) + 3A3 δt 2 cos(x) + 4A2 β2 t 2 cos(x) − 4A2 β2 t 2 cos(2x) + 4A2 γ t 2 sin(2x) − 4Aσ2 t 2 cos(x) + 4Aβ1 t 2 sin(x)))/(8σ1 + 8), θ1 (x, t) = −Aht (cos(x) + Aα sin(2x)), .. . Finally, we have u(x, t) = −A(cos(x) − 1) − (h(4At 2 cos(x) − 3A3 δt 2 cos(3x) + 3A3 δt 2 cos(x) + 4A2 β2 t 2 cos(x) − 4A2 β2 t 2 cos(2x) + 4A2 γ t 2 sin(2x) − 4Aσ2 t 2 cos(x) + 4Aβ1 t 2 sin(x)))/(8σ1 + 8) + · · · , θ (x, t) = −A(cos(x) − 1) − Aht (cos(x) + Aα sin(2x)) + · · · . The solutions given by the HAM contain an auxiliary parameter h, which can be used to control and adjust the convergence region of the series solution. It is interesting that the convergence rate of the approximation series depends on the auxiliary parameter h [18]. The validity region of h is a horizontal line segment. In general, by means of the h-curve, one can choose an acceptable range for h which ensures convergence of the series solution. We have plotted on Figs. 1 and 2 the h-curves of u(0.1, 0.1) and θ(0.1, 0.1) using 4-th order approximate solution. Furthermore, these validity regions confirm the convergence of the obtained series solution. 4.2.

The solution with VIM

In order to apply VIM, we construct the following correction functional, namely  t  1   un,τ (x, τ ) − un,xx (x, τ ) λ1 (τ )(un,τ τ (x, τ ) + un+1 (x, t) = un (x, t) + 1 + σ1 1 + σ1 0 2γ 3δ σ2    un,xx (x, τ ) − un,xx (x, τ ) un,xx (x, τ ) un,x (x, τ ) − un,x2 (x, τ ) + 1 + σ1 1 + σ1 1 + σ1 β1  β2  β2  − un,x (x, τ ) − un,xx (x, τ )) dτ, θn,x (x, τ ) − θn,x (x, τ ) θn (x, τ ) 1 + σ1 1 + σ1 1 + σ1

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t

θn+1 (x, t) = θn (x, t) +

λ2 (τ )(θn,τ (x, τ ) − a un,xτ (x, τ ) − b un,xτ (x, τ ) un,x (x, τ )

0

un,x (x, τ ) θn,xx (x, τ ) − α un,xx (x, τ ) θn,x (x, τ )) dτ, − θn,xx (x, τ ) − α where u˜ n and θ˜n are considered as restricted variations, i.e. δ u˜ n = 0 and δ θ˜n = 0. Making the mentioned above functional stationary, the Lagrange multipliers can be determined as λ1 = τ −t and λ2 = −1. Thus, we get the following iteration formulae  t  1   un,τ (x, τ ) − un,xx (x, τ ) un+1 (x, t) = un (x, t) + (τ − t)(un,τ τ (x, τ ) + 1 + σ1 1 + σ1 0 2γ σ2   un,x (x, τ ) un,xx (x, τ ) − un,xx (x, τ ) + 1 + σ1 1 + σ1 β1  3δ  un,x2 (x, τ ) − θn,x (x, τ ) un,xx (x, τ ) − 1 + σ1 1 + σ1 β2  β2  un,x (x, τ ) − un,xx (x, τ ))dτ, θn,x (x, τ ) θn (x, τ ) − 1 + σ1 1 + σ1  t θn+1 (x, t) = θn (x, t) − (un,τ (x, τ ) + a un,xτ (x, τ ) − b un,xτ (x, τ ) un,x (x, τ ) 0

un,x (x, τ ) θn,xx (x, τ ) − α un,xx (x, τ ) θn,x (x, τ )) dτ, − θn,xx (x, τ ) − α To get the iteration, we start the initial approximations, namely u0 (x, t) = −A(cos(x) − 1), θ0 (x, t) = −A(cos(x) − 1). Using the above iteration formulae and the initial approximation, we obtain u1 (x, t) = −A(cos(x) − 1), θ1 (x, t) = −A(cos(x) − 1), u2 (x, t) = (4At 2 cos(x) + 3A3 δt 2 cos(x) + 4A2 β2 t cos(x) − 3A3 δt 2 cos(3x) − 4A2 β2 t 2 cos(2x) + 4A2 γ t 2 sin(2x) − 4Aσ2 t 2 cos(x) + 4Aβ1 t 2 sin(x))/(8σ1 + 8) − A(cos(x) − 1), . . . θ2 (x, t) = At (cos(x) + Aα sin(2x)) − A(cos(x) − 1), . . . . 4.3.

The solution with the iterative method (He’s polynomials)

Here, we solve the system (1) explicitly using the iterative method (He’s polynomials); the system is equivalent to the following integral equations  t  ζ 1    u(x, t) = −A(cos(x) − 1) + un,τ (x, τ ) + un,xx (x, τ ) − 1 + σ1 1 + σ1 0 0 2γ σ2   un,xx (x, τ ) + un,xx (x, τ ) un,x (x, τ ) − 1 + σ1 1 + σ1

ANALYTIC SOLUTION FOR A NONLINEAR PROBLEM. . .

407

β1  3δ  u2n,x (x, τ ) + θn,x (x, τ ) un,xx (x, τ ) 1 + σ1 1 + σ1  β2  β2  un,x (x, τ ) + un,xx (x, τ ) dτ dζ, θn,x (x, τ ) θn (x, τ ) + 1 + σ1 1 + σ1  t (a un,xτ (x, τ ) + b un,xτ (x, τ ) un,x (x, τ ) θ (x, t) = −A(cos(x) − 1) + +

0

un,x (x, τ ) θn,xx (x, τ ) + α un,xx (x, τ ) θn,x (x, τ )) dτ. + θn,xx (x, τ ) + α Following the algorithm given by Eq. (21), the successive approximations become u0 (x, t) = −A(cos(x) − 1), θ0 (x, t) =−A(cos(x) − 1), u1 (x, t) = (4At 2 cos(x) + 3A3 δt 2 cos(x) + 4A2 β2 t 2 cos(x) − 3A3 δt 2 cos(3x) − 4A2 β2 t 2 cos(2x) + 4A2 γ t 2 sin(2x) − 4Aσ2 t 2 cos(x) + 4Aβ1 t 2 sin(x))/(8σ1 + 8), θ1 (x, t) = At (cos(x) + Aα sin(2x)), .. .. Hence, the series solution of Eq. (1) is given by u(x, t) = (4At 2 cos(x) + 3A3 δt 2 cos(x) + 4A2 β2 t 2 cos(x) − 3A3 δt 2 cos(3x) − 4A2 β2 t 2 cos(2x) + 4A2 γ t 2 sin(2x) − 4Aσ2 t 2 cos(x) + 4Aβ1 t 2 sin(x))/(8σ1 + 8) − A(cos(x) − 1), . . . , θ (x, t) = At (cos(x) + Aα sin(2x)) − A(cos(x) − 1), . . . x

t

uHAM

uVIM

0.1

0.05

5.93252462e-006

5.93252462e-006

5.93252462e-006

0.01

5.03335234e-006

5.03335234e-006

5.03335234e-006

0.1

8.73528494e-006

8.73528493e-006

8.73528493e-006

0.05

1.23264310e-004

1.23264310e-004

1.23264310e-004

0.01

1.22451369e-004

1.22451369e-004

1.22451369e-004

0.5

0.9

uHe s

polynomials

0.1

1.25796952e-004

1.25796952e-004

1.25796952e-004

0.05

3.79013002e-004

3.79013002e-004

3.79013002e-004

0.01

3.78415004e-004

3.78415004e-004

3.78415004e-004

0.1

3.80874541e-004

3.80874541e-004

3.80874541e-004

Table 1. Numerical results of HAM compared to VIM and iterative method for u(x, t).

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A. JAFARIAN et al. x

t

θHAM

θVIM

θNIM

0.1

0.05

5.34911212e-005

5.34911213e-005

5.34911213e-005

0.01

1.48972689e-005

1.48972689e-005

1.48972689e-005

0.1

9.94566885e-005

9.94566898e-005

9.94566898e-005

0.5

0.9

0.05

1.65035010e-004

1.65035011e-004

1.65035011e-004

0.01

1.31149533e-004

1.31149534e-004

1.31149534e-004

0.1

2.05045273e-004

2.05045281e-004

2.05045281e-004

0.05

4.08384626e-004

4.08384626e-004

4.08384626e-004

0.01

3.84570487e-004

3.84570487e-004

3.84570487e-004

0.1

4.36109968e-004

4.36109976e-004

4.36109976e-004

Table 2. Numerical results of HAM compared to VIM and iterative method for θ(x, t). −6

x 10

12 11 10 9 8 7 6 5 −2

−1.5

−1 h

−0.5

0

Fig. 1. The h-curve of u(0.1, 0.1) is given by the 4th-order approximate solution. −6

x 10

12 11 10 9 8 7 6 5 −2

−1.5

−1 h

−0.5

0

Fig. 2. The h-curve of θ (0.1, 0.1) is given by the 4th-order approximate solution.

ANALYTIC SOLUTION FOR A NONLINEAR PROBLEM. . .

409

5.

Results In order to assess accuracy of the mentioned methods and to illustrate these methods in more details, we consider Eq. (1) at nonlinear magneto-thermoelasticity with the following arbitrary constants: a = 0.5, A = 0.001, b = 0.5, a = 1, β1 = β2 = 0.05, γ = 1, δ = 0.8. The importance of HAM is determined by comparing the results with VIM and the iterative method in Tables 1 and 2. On the other hand, it is clear that the approximate solutions using HAM in some first iteration agree with the results obtained by VIM and the iterative method (He’s polynomials). Numerical results clearly display the complete reliability and efficiency of the suggested algorithm. We conclude that these methods are very powerful techniques for solving different kinds of linear and nonlinear problems arising in various fields of science and engineering. Acknowledgement The reviewers’ comments, which have improved the quality of this paper, are greatly appreciated. 6.

Conclusions In this study, a comparative study among HAM, VIM and the iterative method (He’s polynomials) has been used to investigate the coupled harmonic waves nonlinear magneto-thermoelasticity equations under influence of rotation. These methods can be employed to solve magneto-thermoelasticity equation with any boundary/initial conditions. A major difference in analysis between HAM and other methods is that HAM can be used as an appropriate approach for controlling the convergence of the approximation series. The proposed example shows that the results of HAM are in excellent agreement with those obtained by VIM and the iterative method. The results of this study, along with other studies, indicate that these methods can be used to solve a lot of nonlinear problems. This is especially convenient to engineering applications with minimum requirements on calculation and computation. By comparing the other results and the convergence study, we come to the conclusion that the approximate solutions of small order have a good accuracy. Mathematically h can be freely chosen except for zero. However, there is an optimal one which should be determined. For example, in our case, h = 100 does not give the best result. In this paper h is determined by the h-curve, but still the theoretical basis for h-curve is an open and interesting issue. REFERENCES [1] V. K. Agarwal: On electromagnetic-thermoelastic plane waves, Acta Mech. 34 (1979), 181. [2] M. A. Ezzat and M. I. A. Othman: State space approach to generalized magneto-thermoelasticity with thermal relaxation in a medium of perfect conductivity, J. Therm. Stresses 25 (2002), 409. [3] M. I. A. Othman: Lord-Shulman theory under the dependence of the modulus of elasticity on the reference temperature in two dimensional generalized thermoelasticity, J. Therm. Stresses 25 (2002), 1027. [4] C. M. Purushothama: Magnetothermoelastic plane waves, Math. Proc. Camb. Phil. Soc. 61 (1965), 939.

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