A Note on Fractional Electrodynamics. Abstract

Transcription

1 Commun Nonlinear Sci Numer Simula 8 ( A Noe on Fracional lecrodynamics Hosein Nasrolahpour Absrac We invesigae he ime evoluion o he racional elecromagneic waves by using he ime racional Maxwell's equaions. We show ha elecromagneic plane wave has ampliude which exhibis an algebraic decay, a asympoically long imes. Keywords: Fracional Calculus; Fracional lecrodynamics; Fracional Maxwell's quaions PACS: 3.5.De; a. Inroducion Fracional calculus is a very useul ool in describing he evoluion o sysems wih memory, which ypically are dissipaive and o complex sysems. In recen decades racional calculus and in paricular racional dierenial equaions have araced ineres o researches in several areas including mahemaics, physics, chemisry, biology, engineering and economics. Applicaions o racional calculus in he ield o physics and asrophysics have gained considerable populariy and many imporan resuls were obained during he las years [-5]. In classical mechanics, as we can see in Re. [7-9] he racional ormalism leads o relaxaion and oscillaion processes ha exhibi memory and delay. This racional nonlocal ormalism is also applicable on maerials and media ha have elecromagneic memory properies. So he generalied racional Maxwell s equaions can give us new models ha can be used in hese complex sysems. The aim o his work is o invesigae he ime evoluion o he racional elecromagneic waves by using he ime racional Maxwell's equaions. In paricular, we show ha elecromagneic plane wave has an ampliude which exhibis an algebraic decay, a asympoically large imes. For his purpose in he ollowing secion we briely review racional elecrodynamics heory [7, 9].. Fracional elecrodynamics In classical elecrodynamics, behavior o elecric ields (, magneic ields ( B and heir relaions o heir sources, charge densiy ( ( r, and curren densiy ( j ( r,, are described by he ollowing Maxwell s equaions: 4. ( r, ( (. B B (3 c Correspondence: Hosein Nasrolahpour, Babolsar Ciy, Maandaran, Iran. -Mail: hnasrolahpour@gmail.com

2 4 B j ( r, (4 c c Where and are elecric permiiviy and magneic permeabiliy, respecively. Now, inroducing he poenials, vecor A ( x i, and scalar ( x i, B A (5 A (6 c and using he Loren gauge condiion we obain he ollowing decoupled dierenial equaions or he poenials: A( r, 4 A( r, j ( r, (7 c c ( r, 4 (8 ( r, ( r, c where c v and v is he velociy o he wave. Furhermore, or a paricle moion wih charge q in he presence o elecric and magneic ield we can wrie he Loren orce as FL q ( v B (9 here, v is he paricle's velociy. In erms o scalar and vecor poenials, q. (5, 6 we may wrie he Loren orce as A ( FL q ( v ( A c As we can see in Re. [7-9], in classical mechanics, he racional ormalism leads o relaxaion and oscillaion processes ha exhibi memory and delay. This racional nonlocal ormalism is also applicable on maerials and media ha have elecromagneic memory properies. So he generalied racional Maxwell s equaions can give us new models ha can be used in hese complex sysems. Up o now, several dieren kinds o racional elecrodynamics based on he dieren approaches o racional vecor calculus have been invesigaed [-5]. For insance a racional-dimensional space approach o he elecrodynamics is presened [] using he racional Laplacian operaor [7, 8]: D 3 D ( x x x y y y where, hree parameers (, and 3 are used o describe he measure disribuion o space where each one is acing independenly on a single coordinae and he oal dimension o he sysem is D 3. However, in his paper we sudy a new approach on his area [9]. The idea is in ac, o wrie he ordinary dierenial wave equaions in he racional orm wih respec o, by replacing he ime derivaive wih a racional derivaive o order ( namely: 4. ( r, ( (3. B

3 B c (4 4 B j ( r, (5 c c And he q. (5, 6 become B A (6 A (7 c And he Loren orce q. ( becomes A FL q( v ( A (8 c A simple example o applicaion o he q. (8 is provided in Appendix. In above equaions he racional derivaive o order, n n, n N is deined in he Capuo sense: n ( n ( ( d (9 n ( n Where denoes he Gamma uncion. For n, n N he Capuo racional derivaive is deined as he sandard derivaive o order n. Also, noe ha we have inroduced an arbirary quaniy wih dimension o [second] o ensure ha all quaniies have correc dimensions. As we can see rom q. (9 Capuo derivaive describes a memory eec by means o a convoluion beween he ineger order derivaive and a power o ime ha corresponds o inrinsic dissipaion in he sysem. Now we can apply Loren gauge condiion o obain he corresponding ime racional wave equaions or he poenials A( r, j ( r, A( r, 4 ( c c ( r, 4 ( ( r, ( r, ( c I and j, we have he homogeneous racional dierenial equaions A ( r, A( r, ( ( c ( r, (3 ( r, ( c We are ineresed in he analysis o he elecromagneic ields saring rom he equaions. Now we can wrie he racional equaions in ollowing compac orm Z ( x, Z ( x, (4 ( x c where Z ( x, represens boh A ( r, and ( r,. We consider a polaried elecromagneic wave, hen A x, A y, A. A paricular soluion o his equaion may be ound in he orm (

4 ikx Z ( x, Z e u ( where k is he wave vecor in he x direcion and Z is a consan. Subsiuing q. (5 ino q. (4 we obain d u ( (6 ( u d where ( ( (7 v k and is he undamenal requency o he elecromagneic wave. Using he Laplace inegral ransormaions, one obains he soluions: u ( u ( ( (8 or he case o and u ( u ( ( u ( (, or he case o,where u ( u (. So wih he boundary condiions u ( u and u ( u (3 he general soluion o he q. (6 may be a u ( u ( or (3 b u ( u ( u,( or where k ( (3 k ( k is one-parameer Miag-Leler uncion. Subsiuing he q. (3a in q. (5 we have a paricular soluion o he equaion as ikx Z ( x, ( Z u e ( (33 We can easily see ha in he case, he soluion o he equaion is ( Z ( x, Re( Z i kx u e (34 which deines a periodic, wih undamenal period T, monochromaic wave in he, x, direcion and in ime,.this resul is very well known rom he ordinary elecromagneic waves heory. However or he arbirary case o ( he soluion is periodic only respec o x and i is no periodic wih respec o.the soluion represens a plane wave wih ime decaying ampliude. For example or he case we have u ( ( e where or simpliciy, we have used u iniial condiion. Thereore he soluion is Z x Z e e (, ( ikx (5 (9 (35 (36

5 Also or he case o we have u( ( e ( er ( e erc ( where erc denoes he complimenary error uncion and he error uncion is deined as er ( e d, erc ( er (, C For large values o,he complimenary error uncion can be approximaed as erc ( exp( Subsiuing q. (37 ino q. (5 leads o he soluion 3 (, ( 3 4 ( ikx Z x Z e erc e (4 A asympoically large imes, and using q. (39 we have Z ikx Z ( x, ( e (4 3 Then or hese cases, he soluions are periodic only respec o x and hey are no periodic wih respec o. In ac soluions represen plane waves wih ime decaying ampliude. 3. Asympoic behavior o he soluion The algebraic decay o he soluions o he racional equaions is he mos imporan eec o he racional derivaive in he ypical racional equaions conrary o he exponenial decay o he usual sandard orm o he equaions. To describe his algebraic decay in our case, we consider he inegral orm or he Miag-Leler uncion. The asympoic expansion o ( based on he inegral represenaion o he Miag-Leler uncion in he orm [6] exp( (4 ( d i where (, (, C and he pah o inegraion is a loop saring and ending a and encircling he circular disk in he posiive sense arg on. The inegrand has a branch poin a.the complex -plane is cu along he negaive real axis and in he cu plane he inegrand is single-valued he principal branch o is aken in he cu plane. q. (4 can be proved by expanding he inegrand in powers o and inegraing erm by erm by making use o he well-known Hankel s inegral or he reciprocal o he gamma uncion, namely e d ( i (43 The inegral represenaion q. (4 can be used o obain he asympoic expansion o he Miag-Leler uncion a ininiy. Accordingly, he ollowing cases are obained. I and is a real number such ha min[, ] hen or N N, N here holds he ollowing asympoic expansion: H a (37 (38 (39

6 as, arg ( N ( exp( O[ ] r N r ( r and N ( O[ ] r N r ( r as, arg. In our case, and ( ( Then, subsiuion o q. (46 ino q. (33 gives Z ikx (47 Z ( x, [( ] e ( As we can see in his resul, we arrive o he asympoic soluion or he elecromagneic wave equaion which represens a plane wave wih algebraic ime-decaying ampliude. This is a direc consequence o he racional ime derivaive in he sysem. In he oher word racional diereniaion wih respec o ime can be inerpreed as an exisence o memory eecs which correspond o inrinsic dissipaion in our sysem. (44 (45 (46 4. Conclusion The asympoic behavior o Miag-Leler uncions [6] plays a very imporan role in he inerpreaion and undersanding o he soluions o various problems o physics conneced wih racional phenomena ha occur in complex sysems. In his aricle we have sudied he ime evoluion o he racional elecromagneic waves by using he ime racional Maxwell's equaions. We showed ha elecromagneic plane wave has ampliude which exhibis an algebraic decay or in our case (q. (4, 47. Appendix: Fracional dynamics o charged paricles For he simples case we can consider moion o charged paricles in a uniorm elecric ield kˆ. So using he racional Newon s second law we have c D p ( q (48 i p (, so we have q p ( ( (49 where p is he -componen o paricle's momenum. Also we can easily calculae -componen o paricle's posiion as a uncion o ime, i.e. ( rom

7 m p ( ( D ( c (5 Taking ino accoun he iniial condiion ( and subsiuing q. (49 ino q. (5 leads o he soluion q ( m( (5 For he case o we can easily show ha ( q q m(3 m as expeced rom he sandard elecrodynamics. (5 Reerences [] R. Hiler, Applicaions o Fracional Calculus in Physics (World Scieniic Press,. [] J. Sabaier e al. (ds., Advances in Fracional Calculus (Springer, 7. [3] J. A. Tenreiro Machado, e al. (ds., Nonlinear and Complex Dynamics (Springer,. [4] J. Klaer e al. (ds., Fracional Dynamics: Recen Advances (World Scieniic,. [5] R. Herrmann, Fracional Calculus (World Scieniic Press,. [6] V.. Tarasov, Fracional Dynamics (Springer, HP,. [7] H. Nasrolahpour, Prespaceime J. 3 ( ( [8] H. Nasrolahpour, Prespaceime J. 3 ( ( [9] F. Mainardi, Chaos Sol. Frac. 7(9 ( [] H. Nasrolahpour, Prespaceime J. 3 (3 ( [] J. F. Gome-Aguilar e al., Rev. Mex. Fis. 58 ( [] L. Vaque, Frac. Calc. Appl. Anal. 4 (3 ( [3] A. A. Sanislavsky, Asrophysical Applicaions o Fracional Calculus (Springer,. [4] J. An e al., Mon. No. R. Asron. Soc. 4 ( [5] V. V. Uchaikin, J. xp. Theor. Phys. Le., 9(4 ( 5. [6] H. J. Haubold e al., J. Appl. Mah. ( [7] F. H. Sillinger, J. Mah. Phys. 8(6 ( [8] C. Palmer, P.N. Savrinou, J. Phys. A 37 ( [9] J. F. Gome e al., prin: mah-ph / [] M. Zubair e al., Nonlinear Anal.: Real World Appl. (5 ( [] N. nghea, Microwave and Op. Technol. Le.7 ( ( [] Q. A. Naqvi, M. Abbas, Op. Commun. 4 ( [3] V.. Tarasov, Phys. Plasmas (5 86. [4] A. Hussain, Q.A. Naqvi, Prog. lecromagn. Res. 59 ( [5] V.. Tarasov, Ann. Phys. 33 (