homotopy analysis method (ham) is applied to the maxwell system

8/2/2019 Homotopy Analysis Method (HAM) is Applied to the Maxwell System

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The Journal of Mathematics and Computer Science Vol .3 No.2 (2011) 225 - 235

Numerical solution for Maxwell's equation in metamaterials by

Homotopy Analysis Method

1. Malek Ashtar University of Technology

2. Department of Mathematics, Shahid Beheshti University, Tehran, Iran

[email protected]

[email protected]

Received: May 2011, Revised: June 2011

Online Publication: December 2011

Abstract

In this paper, the Homotopy analysis Method (HAM) is applied to the Maxwell system.

The HAM yields an analytical solution in terms of a rapidly convergent infinite power

series with easily computable terms.

1. Basic of Homotopy analysis method

We consider the following differential equation

,0)]([ uN (1)

where N is a nonlinear operator, denotes independent variable, )(u is an

unknown function, respectively. For simplicity, we ignore all boundary or initial

conditions, which can be treated in the similar way. By means of generalizing the

traditional homotopy method, Liao [7] constructs the so-called zero-order

deformation equation

)],;([)()]();([)1( 0 pNphHupLp (2)

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Where 1][0,p is the embedding parameter, 0h is a non-zero auxiliary parameter,

)(H

is an auxiliary function, L is an auxiliary linear operator, )(0 u is an initial guess of is

an

unknown function, respectively. It is important, that one has great freedom to choose

auxiliary things in HAM. Obviously, when = 0 = 1, it holds)()0;( 0 u , )()1;( u (3)

respectively. Thus, as p increases from 0 to 1, the solution );( p varies from the initial

guess )(0 u to the solution )(u .Expanding );( p in Taylor series with respect to p ,

we have

1

0 ,)()();(m

m

m puup (4)

Where

.);(

!

1)(

0

p

m

m

mp

p

mu

(5)

If the auxiliary linear operator, the initial guess, the auxiliary parameter h , and the

auxiliary function are properly chosen, the series (4) converges at 1p , and then we

have

,)()()(1

0

m

muuu (6)

which must be one of solutions of original nonlinear equation, as proved by Liao [1]. As

1h and ,1)( H Eq. (2) become

0)];([)]();([)1( 0 ppNupLp (7)

Which is used mostly in the homotopy perturbation method, where as the solution

obtained directly, without using Taylor series [2, 3]. According to the definition (5), the

governing equation can be deduced from the zero-order deformation equation (2).

Define the vector

)}.(,),(),({ 10 nn uuuu

Differentiating equation (7) m times with respect to the embedding parameter p and

then setting 0p and finally dividing them by m!, we have the so-called m th-order

deformation equation

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),()()]()([ 11

mmmmm uRhHuuL (8)

subject to initial condition

,0)0,( xum ,0

)0,(

t

xum (9)

Where

.)];([

)!1(

1)(

0

1

1

1

p

m

m

mmp

pN

muR

(10)

And

m =

1

0

1

1

m

m(11)

It should be emphasized that )(mu for 1m is governed by the linear equation (8)

with the linear boundary conditions that come from original problem, which can be

easily solved by symbolic computation software such as Mathematica. If Eq. (1) admits

unique solution, then this method will produce the unique solution. If equation (1) does

not possess unique solution, the HAM will give a solution among many other (possible)

solutions. For the convergence of the above method we refer the reader to [5,6, 7, 9 ].

2. The Maxwell system in meta materials

The DNG, metamaterials can be simulated using lossy Drude polarization and

magnetization modes. The governing equations for modeling wave propagation in

metamaterials are[1,4,10]

0

=

12

0 = . 1310 2 + 02 = , 1410 2 + 02 = , 15

where 0 is the vacuum permittivity,0 is the vacuum permeability, and arethe electric and magnetic plasma frequencies, respectively, and are the electric and



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magnetic damping frequencies, respectively,(, ) and(, ) are the electric andmagnetic fields, respectively, and(, )and (, )are the induced electric andmagnetic currents, respectively. The complex materials make solving the metamaterials

model more challenging, since the governing equations cannot be reduced to a simple

vector wave equation as in vacuum.

For simplicity, we assume that the modeling domain be (0,), where is abounded Lipschitz polyhedral domain in 3 with connected boundary. Furthermore,we assume that the boundary of is perfect conducting so that

= 0 On (16)Where is the unit outward normal to . Also we assume that the initial conditionsare

, 0 = 0, , 0 = 0 (17), 0 =0,, 0 =0.Lemma 2.1. There exists a unique solution for system (12)-(15). Furthermore; the

solution of the system (12)-(15) satisfies the following stability estimate [4]

0()02 + 0 ()02 + 10 2 ()02 + 10 2 ()02 0(0)02 +

0

(0)

0

2+

1

0

2

(0)

0

2+

1

0

2

(0)

0

2. (18)

Lemma 2.2. Assume that the initial conditions are divergence free.i.e.,. 00 = 0, . 00 = 0. 0 = 0, . 0 = 0 (19)Then for any time > 0, the electric field and electric current are divergence free.Similary, for any time > 0, the magnetic field and the magnetic currentaredivergence free.

3. The HAM for the Maxwell system in meta materials

In this section of the paper, we consider the Maxwell system in Meta materials. In

system (12)-(15), we carry directly to 2D by using the scalar and vector curl operators

= 21 1 2 (20) = = ( 2 , 1) (21)

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Where the upper index denotes the vector transpose. Considering the lossy propertyof metamaterials, we construct the following exact solution for the 2D transverse

electrical model (assuming all physical parameters to be one, i.e.(0 = 0 = =

=

=

= 1)

= = 12 cos sin

12 sin cos etcost

= 2 (cos cos) etcost(22)

= = 12 cos sin12 sin cos

etsin t

= 2 (cos cos) et sintTo solve the system. (1215) by means of homotopy analysis method, we choose the

linear operator

L1[1x,y,t;p] = 1x,y,t;pt L22x,y,t;p = 2x,y,t;pt L33x,y,t;p = 3x,y,t;pt (23)L4[

4

x,y,t;p

] =

4

x,y,t;p

t

L5[5x,y,t;p] = 5x,y,t;pt L6[6x,y,t;p] = 6x,y,t;pt

With the property: = 0 = 1,2,3,4,5,6where

are integral constants. The inverse L

i1operator are given by

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Li1

= . dtt0

We now define a nonlinear operator as

11,2,3 ,4 ,5 ,6 = 0 1t 3y + 4 21,2,3 ,4 ,5 ,6 = 0 2t + 3x + 531,2 ,3 ,4,5,6 = 0 3t + 2x 1y 6

41,

2,

3,

4,

5,

6=

1

02 4

t+

02 4 1 (24)

51,2,3 ,4 ,5,6 = 102 5t + 02 5 261,2 ,3,4,5 ,6 = 1

02 6t + 02 6 3

Using above definition, we construct the zeroth-order deformation equation

1 pL11x,y,t;p Ex 0x,y,t = 11[1,2,3 ,4,5,6]1 pL2 2x,y,t;p Ey 0x,y,t = 22[1,2 ,3,4,5 ,6]1 pL33x,y,t;p H0x,y,t = 33[1,2,3,4 ,5,6] (25)1 pL44x,y,t;p Jx 0x,y,t = 44[1,2 ,3,4,5,6]1

p

L

5 5x,y,t;p

Jy 0

x,y,t

=

55[

1,

2,

3,

4,

5,

6]

1 pL66x,y,t;p K0x,y,t = 66[1,2,3,4,5,6]With the initial condition:

1x,y,t;0 = Ex 0x,y,t 1x,y,t;1 = Exx,y,t

2

x,y,t;0

= Ey 0

x,y,t

2

x,y,t;1

= Ey

x,y,t

3x,y,t;0 = 0x,y,t 3x,y,t;1 = Hx,y,t



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4x,y,t;0 = Jx x,y,t 4x,y,t;1 = Jxx,y,t 5x,y,t;0 = Jy x,y,t 5x,y,t;1 = Jyx,y,t x,y,t;0 = K x,y,t x,y,t;1 = Kx,y,t Thus, we obtain the th-order deformation equations

L1 Ex mx,y,t m Ex m1x,y,t = 11 [Ex m1, Ey m1, m1, Jx m1, Jy m1, m1]L2 Ey mx,y,t m Ey m1x,y,t = 22 [Ex m1, Ey m1 , m1, Jx m1 , Jy m1, m1]

L3 Hm x,y,t m Hm1x,y,t = 33 [Ex m1, Ey m1, m1, Jx m1, Jy m1, m1] (26)L4 Jx mx,y,t m Jx m1x,y,t = 54 [Ex m1 , Ey m1, m1, Jx m1, Jy m1 , m1]L5 Jy mx,y,t m Jy m1x,y,t = 55 [Ex m1 , Ey m1, m1, Jx m1, Jy m1 , m1]L6 Hmx,y,t m Hm1x,y,t = 66 [Ex m1, Ey m1 , m1, Jx m1 , Jy m1, m1]

where

1

Ex

m

1

, Ey

m

1

,

m

1, Jx

m

1

, Jy

m

1

,

m

1

= 0

Ex m1

t

Hm1

y

+ Jx m

1

2 Ex m1, Ey m1 , m1 , Jx m1, Jy m1 , m1 = 0 Ey m1t + Hm1x + Jy m13 Ex m1, Ey m1, m1, Jx m1 , Jy m1 , m1 = 0 Hm1t + Ey m1x Ex m1y m1 (27)

4 Ex m1, Ey m1, m1 , Jx m1, Jy m1, m1 = 10 2 Jx m1t + 0 2 Jx m1 Ex m1

5 Ex m1, Ey m1 , m1, Jx m1, Jy m1, m1 =1

02 Jy

m1t + 02 Jx m1 Ey m16 Ex m1 , Ey m1, m1 , Jx m1 , Jy m1, m1 = 102 m1t + 02 m1 Hm1

Now the solution of the -th-order deformation Eq. (26) 1Ex mx,y,t = m Ex m1x,y,t + 11[1 [Ex m1, Ey m1 , m1 , Jx m1, Jy m1, m1]Ey

mx,y,t

= m Ex

m1x,y,t

+

2

1[

2

[Ex

m1, Ey

m1,

m

1 , Jx

m1, Jy

m1,

m

1]



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Hmx,y,t = m Hm1x,y,t + 31[3 Ex m1, Ey m1, m1, Jx m1 , Jy m1 , m1 (28)Jx mx,y,t = m Jx m1x,y,t + 41[4 [Ex m1, Ey m1 , m1, Jx m1 , Jy m1, m1]Jy mx,y,t = m Jx m1x,y,t + 5

1

[5 [Ex m1, Ey m1 , m1, Jx m1 , Jy m1, m1]Kmx,y,t = m Km1x,y,t + 61[3 [Ex m1 , Ey m1, m1 , Jx m1 , Jy m1, m1]We start with under initial approximations:

By means of the above iteration formula (28) ifhi = 1, we can obtain directly theother components asEx,y,t = ExEy = Ex 0x,y,t + Ex x,y,t +i=0y0x,y,t + y ix,y,t +i=0 = cos sin (cos ()+42 1+22 1 22 +

cos sin (cos ()+422+22 2 2

2

+

Hx,y,t = H0x,y,t + Hix,y,t +

i=0

=

2

cos

cos

(cos (

) + 2

3 +

3

2+



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, , = Jx

y = Jx0x,y,t + Jx x,y,t +i=0

y0

x,y,t

+

y i

x,y,t

+i=0

=

t cos

sin

4(

2

2sin (

)+2

42+

2

42+

2

42

)

222 + t cos sin 5(22sin ()+2 5 2+ 2 52+ 25 2)222 +

x,y,t = K0x,y,t + Kix,y,t

+

i=0

= t cos cos6 (2 sin (t)2 + 262 + 262 + 26222

+ We now choose: = , = , = , = , = , = , = , = ,and then obtain:

x,y,t = t cos cos6 (4 2 + 26 + 62 2

And then choose: = , = , = , = , = , = , = , = ,with this chose, we have:



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x,y,t = t cos cos (1560 + 516 + 442 + 3)96

2

Conclusion

Homotopy analysis method is a powerful method which yields a convergent series

solution for Maxwell's equation system. This method is better than numerical methods,

as it is free from rounding off errors, and does not require large computer power.

It is apparently seen that HAM is a very powerful and efficient technique in finding

analytical solutions for wide classes of linear and nonlinear problems. The results show

that HAM is a powerful mathematical tool for solving nonlinear partial differential

equations and systems of nonlinear partial differential equations.

Mathematica has been used for computations in this article.

References

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Mathemathics,And Compution, 77(1996)133-135.

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[3] He JH. Some asymptotic methods for strongly nonlinear equations, Int J Mod Phys

B, 20(10) (2006) 1141-1199.

[4]Jichun Li, Numerical convergence and physical fidelity analysis for Maxwell'sequations in metamaterials,comput.Methods Appl.Mech.Engrg.198(2009)3161-3172.

[5]Jafari H., Saeidy M., Firoozjaee M. A.,The Homotopy Analysis Method for SolvingHigher Dimensional Initial Boundary Value Problems of Variable Coefficients, Numerical

Methods for Partial Differential Equations, Wiley Periodicals, Inc. Numer

Methods Partial Differential Eq 26(2010)10211032.



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[6] Jafari H., FiroozjaeeM. A. Multistage Homotopy Analysis Method for Solving

Nonlinear Integral Equations, Applications and Applied Mathematic International

Journal

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[7] Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. CRC

Press, Boca Raton:Chapman & Hall; 2003.

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homotopy analysis method (ham) is applied to the maxwell system

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