Commun Nonlinear Sci Numer Simulat 18 (2013) 2589–2593

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

A note on fractional electrodynamics

1007-5704/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cnsns.2013.01.005

E-mail address: hnasrolahpour@gmail.com

Hosein NasrolahpourUniversity Town, P.O. Box 47419–73647, Babolsar, Mazandaran, Iran

a r t i c l e i n f o

Article history:Received 3 August 2012Accepted 9 January 2013Available online 25 January 2013

Keywords:Fractional calculusFractional electrodynamicsFractional Maxwell’s equations

a b s t r a c t

We investigate the time evolution of the fractional electromagnetic waves by using thetime fractional Maxwell’s equations. We show that electromagnetic plane wave has ampli-tude which exhibits an algebraic decay, at asymptotically large times.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Fractional calculus is a very useful tool in describing the evolution of systems with memory, which typically are dissipativeand to complex systems. In recent decades fractional calculus and in particular fractional differential equations have attractedinterest of researches in several areas including mathematics, physics, chemistry, biology, engineering and economics. Appli-cations of fractional calculus in the field of physics and astrophysics have gained considerable popularity and many importantresults were obtained during the last years [1–15]. In classical mechanics, as we can see in Refs. [7–9], the fractional formalismleads to relaxation and oscillation processes that exhibit memory and delay. This fractional nonlocal formalism is alsoapplicable on materials and media that have electromagnetic memory properties. So the generalized fractional Maxwell’sequations can give us new models that can be used in these complex systems. The aim of this work is to investigate the timeevolution of the fractional electromagnetic waves by using the time fractional Maxwell’s equations. In particular, we showthat electromagnetic plane wave has an amplitude which exhibits an algebraic decay, at asymptotically large times. For thispurpose in the following section we briefly review fractional electrodynamics theory [7, 17].

2. Fractional electrodynamics

In classical electrodynamics, behavior of electric fields (~E), magnetic fields (~B) and their relations to their sources, chargedensity (qð~r; tÞ) and current density (~jð~r; tÞ), are described by the following Maxwell’s equations:

~r �~E ¼ 4pe

qð~r; tÞ ð1Þ

~r �~B ¼ 0 ð2Þ

~r�~E ¼ �1c@~B@t

ð3Þ

2590 H. Nasrolahpour / Commun Nonlinear Sci Numer Simulat 18 (2013) 2589–2593

~r�~B ¼ 4plc~jð~r; tÞ þ el

c@~E@t

ð4Þ

where e and l are electric permittivity and magnetic permeability, respectively. Now, introducing the potentials, vector~Aðxi; tÞ and scalar u(xi, t)

~B ¼ ~r�~A ð5Þ

~E ¼ �1c@~A@t� ~ru ð6Þ

and using the Lorenz gauge condition we obtain the following decoupled differential equations for the potentials:

D~Að~r; tÞ � elc2

@2~Að~r; tÞ@t2 ¼ �4p

c~jð~r; tÞ ð7Þ

Duð~r; tÞ � elc2

@2uð~r; tÞ@t2 ¼ �4p

eqð~r; tÞ ð8Þ

where elc2 ¼ 1

v2 and v is the velocity of the wave. Furthermore, for a particle motion with charge q in the presence of electricand magnetic field we can write the Lorentz force as

~FL ¼ qð~Eþ~v �~BÞ ð9Þ

here, v is the particle’s velocity. In terms of scalar and vector potentials, Eqs. (5), and (6) we may write the Lorentz force as

~FL ¼ q �1c@~A@t� ~ruþ~v � ð~r�~AÞ

!ð10Þ

As we can see in Refs. [7–9], in classical mechanics, the fractional formalism leads to relaxation and oscillation processesthat exhibit memory and delay. This fractional nonlocal formalism is also applicable on materials and media that have elec-tromagnetic memory properties. So the generalized fractional Maxwell’s equations can give us new models that can be usedin these complex systems. Up to now, several different kinds of fractional electrodynamics based on the different approachesto fractional vector calculus have been investigated [18–23]. However, in this paper we study a new approach on this area[17]. The idea is in fact, to write the ordinary differential wave equations in the fractional form with respect to t, by replacingthe time derivative with a fractional derivative of order a (0 < a 6 1) namely:

~r �~E ¼ 4pe

qð~r; tÞ ð11Þ

~r �~B ¼ 0 ð12Þ

~r�~E ¼ �1c

1g1�a

@a~B@ta

ð13Þ

~r�~B ¼ 4plc~jð~r; tÞ þ el

c1

g1�a

@a~E@ta ð14Þ

and the Eqs. (5) and (6) become

~B ¼ ~r�~A ð15Þ

~E ¼ � 1cg1�a

@a~A@at� ~ru ð16Þ

and the Lorentz force Eq. (10) becomes

~FL ¼ q � 1cg1�a

@a~A@at� ~ruþ~v � ð~r�~AÞ

!ð17Þ

A simple example of application of the Eq. (17) is provided in Appendix In above equations the fractional derivative oforder a, n � 1 < a < n, n 2 N is defined in the Caputo sense:

@af ðtÞ@ta

¼ 1Cðn� aÞ

Z t

0ðt � sÞn�a�1 @

nf ðsÞ@sn

ds ð18Þ

H. Nasrolahpour / Commun Nonlinear Sci Numer Simulat 18 (2013) 2589–2593 2591

where C denotes the Gamma function. For a = n, n 2 N the Caputo fractional derivative is defined as the standard derivativeof order n. Also, note that we have introduced an arbitrary quantity g with dimension of [second] to ensure that all quantitieshave correct dimensions. As we can see from Eq. (18) Caputo derivative describes a memory effect by means of a convolutionbetween the integer order derivative and a power of time that corresponds to intrinsic dissipation in the system. Now we canapply Lorentz gauge condition to obtain the corresponding time fractional wave equations for the potentials.

D~Að~r; tÞ � elc2

1g2ð1�aÞ

@2a~Að~r; tÞ@t2a ¼ �4p

c~jð~r; tÞ ð19Þ

Duð~r; tÞ � elc2

1g2ð1�aÞ

@2auð~r; tÞ@t2a ¼ �4p

eqð~r; tÞ ð20Þ

If q = 0 and~j ¼ 0, we have the homogeneous fractional differential equations

D~Að~r; tÞ � elc2

1g2ð1�aÞ

@2a~Að~r; tÞ@t2a ¼ 0 ð21Þ

Duð~r; tÞ � elc2

1g2ð1�aÞ

@2auð~r; tÞ@t2a ¼ 0 ð22Þ

We are interested in the analysis of the electromagnetic fields starting from the equations. Now we can write the frac-tional equations in following compact form

@2Zðx; tÞ@x2 � el

c2

1g2ð1�aÞ

@2aZðx; tÞ@t2a ¼ 0 ð23Þ

where Z(x, t) represents both ~Að~r; tÞ and uð~r; tÞ. We consider a polarized electromagnetic wave, then Ax = 0, Ay – 0, Az – 0. Aparticular solution of this equation may be found in the form

Zðx; tÞ ¼ Z0e�ikxuðtÞ ð24Þ

where k is the wave vector in the x direction and Z0 is a constant. Substituting Eq. (24) into Eq. (23) we obtain

d2auðtÞdt2a þX2

f uðtÞ ¼ 0 ð25Þ

where

X2f ¼ v2k2g2ð1�aÞ ¼ X2g2ð1�aÞ ð26Þ

and X is the fundamental frequency of the electromagnetic wave. The solution of this equation may be

uðtÞ ¼ E2að�X2f t2aÞ ð27Þ

where

EaðzÞ ¼X1k¼0

zk

Cð1þ akÞ ð28Þ

is one-parameter Mittag–Leffler function. Substituting the Eq. (27) in Eq. (24) we have a particular solution of the equa-tion as

Zðx; tÞ ¼ Z0e�ikxE2að�X2f t2aÞ ð29Þ

We can easily see that in the case a = 1, the solution to the equation is

Zðx; tÞ ¼ ReðZ0eiðXt�kxÞÞ ð30Þ

which defines a periodic, with fundamental period T = 2pX, monochromatic wave in the, x, direction and in time, t. This re-sult is very well known from the ordinary electromagnetic waves theory. However for the arbitrary case of a(0 < a < 1) thesolution is periodic only respect to x and it is not periodic with respect to t. The solution represents a plane wave with timedecaying amplitude. For example for the case a ¼ 1

2 we have

uðtÞ ¼ E1ð�gX2tÞ ¼ e�gX2t ð31Þ

Therefore the solution is

Zðx; tÞ ¼ ðZ0e�gX2tÞe�ikx ð32Þ

2592 H. Nasrolahpour / Commun Nonlinear Sci Numer Simulat 18 (2013) 2589–2593

Also for the case of a ¼ 14 we have

uðtÞ ¼ E12�g3

2X2 ffiffitp� �

¼ eg3X4t 1þ erf ð�g32X2 ffiffi

tpÞ

� �¼ eg3X4terfc g3

2X2 ffiffitp� �

ð33Þ

where erfc denotes the complimentary error function and the error function is defined as

erf ðzÞ ¼ 2ffiffiffiffipp

Z z

0e�t2

dt; erfcðzÞ ¼ 1� erf ðzÞ; z 2 C ð34Þ

For large values of z, the complimentary error function can be approximated as

erfcðzÞ ’ 1ffiffiffiffipp

zexpð�z2Þ ð35Þ

Substituting Eq. (33) into Eq. (24) leads to the solution

Zðx; tÞ ¼ Z0eg3X4terfc g32X2 ffiffi

tp� �� �

e�ikx ð36Þ

At asymptotically large times, and using Eq. (35) we have

Zðx; tÞ ’ Z0ffiffiffiffipp

g32X2 ffiffi

tp

!e�ikx ð37Þ

Then for these cases, the solutions are periodic only respect to x and they are not periodic with respect to t. In fact solu-tions represent plane waves with time decaying amplitude.

3. Asymptotic behavior of the solution

The algebraic decay of the solutions of the fractional equations is the most important effect of the fractional derivative inthe typical fractional equations contrary to the exponential decay of the usual standard form of the equations. To describethis algebraic decay in our case, we consider the integral form for the Mittag–Leffler function. The asymptotic expansion ofEa(z) based on the integral representation of the Mittag–Leffler function in the form [16]

EaðzÞ ¼1

2pi

Z!

ta�1 expðtÞta � z

dt ð38Þ

where R(a) > 0, (a,z) 2 C and the path of integration Y is a loop starting and ending at �1 and encircling the circular diskjtj 6 jzj

1a in the positive sense |arg t| < p on Y. The integrand has a branch point at t = 0 .The complex t-plane is cut along

the negative real axis and in the cut plane the integrand is single-valued the principal branch of ta is taken in the cut plane.Eq. (38) can be proved by expanding the integrand in powers of t and integrating term by term by making use of the well-known Hankel’s integral for the reciprocal of the gamma function, namely

1CðbÞ ¼

12pi

ZHa

ef

fb df ð39Þ

The integral representation Eq. (38) can be used to obtain the asymptotic expansion of the Mittag–Leffler function atinfinity. Accordingly, the following cases are obtained.

If 0 < a < 2 and l is a real number such that pa2 < l < min½p;pa� then for N⁄ 2 N, N⁄– 1 there holds the following asymp-

totic expansion:

EaðzÞ ¼1a

zð1�bÞ

a expðz1aÞ �

XN�r¼1

1Cð1� arÞ

1zr þ O

1zN�þ1

� �ð40Þ

as |z| ?1, j arg zj 6 l and

EaðzÞ ¼ �XN�r¼1

1Cð1� arÞ

1zr þ O

1zN�þ1

� �ð41Þ

as |z| ?1, l 6 j arg zj 6 p. In our case, z ¼ �X2f t2a and

E2a �X2f t2a

� �’ 1

Cð1� 2aÞ1

X2f t2a

ð42Þ

Then, substitution of Eq. (42) into Eq. (29) gives

Zðx; tÞ ’ ð Z0

X2f Cð1� 2aÞ

Þt�2a

" #e�ikx ð43Þ

H. Nasrolahpour / Commun Nonlinear Sci Numer Simulat 18 (2013) 2589–2593 2593

As we can see in this result, we arrive to the asymptotic solution for the electromagnetic wave equation which representsa plane wave with algebraic time-decaying amplitude. This is a direct consequence of the fractional time derivative in thesystem. In the other word fractional differentiation with respect to time can be interpreted as an existence of memory effectswhich correspond to intrinsic dissipation in our system.

4. Conclusion

The asymptotic behavior of Mittag–Leffler functions [16] plays a very important role in the interpretation and under-standing of the solutions of various problems of physics connected with fractional phenomena that occur in complex sys-tems. In this article we have studied the time evolution of the fractional electromagnetic waves by using the timefractional Maxwell’s equations. We showed that electromagnetic plane wave has amplitude which exhibits an algebraic de-cay for t ?1 in our case (Eq. (37) and (43)).

Appendix A. Fractional dynamics of charged particles

For the simplest case we can consider motion of charged particles in a uniform electric field ~E ¼ Ezk̂. So using the frac-tional Newton’s second law we have

0cDa

t pzðtÞ ¼ g1�aqEz ð44Þ

Pz(0) = 0, so we have

pzðtÞ ¼qEzg1�a

Cðaþ 1Þ ta ð45Þ

where pz is the z-component of particle’s momentum. Also we can easily calculate z-component of particle’s position as afunction of time, i.e. z(t) from

pzðtÞ ¼m

g1�a ð0cDa

t zðtÞÞ ð46Þ

Taking into account the initial condition z(0) = 0 and substituting Eq. (45) into Eq. (46) leads to the solution

zðtÞ ¼ qEzg2�2a

mCð2aþ 1Þ t2a ð47Þ

For the case of a = 1 we can easily show that

za¼1ðtÞ ¼qEz

mCð3Þ t2 ¼ qEz

2mt2 ð48Þ

as expected from the standard electrodynamics.

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