PIER
 
Progress In Electromagnetics Research
ISSN: 1070-4698, E-ISSN: 1559-8985
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 114 > pp. 255-269

DIFFERENTIAL ELECTROMAGNETIC EQUATIONS IN FRACTIONAL SPACE

By M. Zubair, M. J. Mughal, Q. A. Naqvi, and A. A. Rizvi

Full Article PDF (158 KB)

Abstract:
The present study deals with a novel approach for fractional space generalization of the differential electromagnetic equations. These equations can describe the behavior of electric and magnetic fields in any fractal media. A new form of vector differential operator Del, and its related differential operators, is formulated in fractional space. Using these modified vector differential operators, the classical Maxwell equations have been worked out for fractal media. The Laplace, Poisson and Helmholtz equations in fractional space are derived by using modified vector differential operators. Also a new fractional space generalization of the potentials for static and time varying fields is presented.

Citation:
M. Zubair, M. J. Mughal, Q. A. Naqvi, and A. A. Rizvi, "Differential Electromagnetic Equations in Fractional Space," Progress In Electromagnetics Research, Vol. 114, 255-269, 2011.
doi:10.2528/PIER11011403
http://www.jpier.org/PIER/pier.php?paper=11011403

References:
1. Stillinger, F. H., "Axiomatic basis for spaces with noninteger dimension," J. Math. Phys., Vol. 18, No. 6, 1224-1234, 1977.
doi:10.1063/1.523395

2. He, X., "Anisotropy and isotropy: A model of fraction dimensional space," Solid State Commun., Vol. 75, 111-114, 1990.
doi:10.1016/0038-1098(90)90352-C

3. Palmer, C. and P. N. Stavrinou, "Equations of motion in a noninteger-dimension space," J. Phys. A, Vol. 37, 6987-7003, 2004.
doi:10.1088/0305-4470/37/27/009

4. Willson, K. G., "Quantum field-theory, models in less than 4 dimensions," Phys. Rev., Vol. 7, No. 10, 2911-2926, 1973.

5. Bollini, C. G. and J. J. Giambiagi, "Dimensional renormalization: The number of dimensions as a regularizing parameter," Nuovo Cimento B, Vol. 12, 20-26, 1972.

6. Ashmore, J. F., "On renormalization and complex space-time dimensions," Commun. Math. Phys., Vol. 29, 177-187, 1973.
doi:10.1007/BF01645246

7. Agrawal, O. P., "Formulation of Euler-Lagrange equations for fractional variational problems," J. Math. Anal. Appl., Vol. 271, No. 1, 368-379, 2002.
doi:10.1016/S0022-247X(02)00180-4

8. Baleanu, D. and S. Muslih, "Lagrangian formulation of classical ¯elds within Riemann-Liouville fractional derivatives," Phys. Scripta, Vol. 72, No. 23, 119-121, 2005.
doi:10.1238/Physica.Regular.072a00119

9. Tarasov, V. E., "Electromagnetic fields on fractals," Modern Phys. Lett., Vol. 21, No. 20, 1587-1600, 2006.
doi:10.1142/S0217732306020974

10. Tarasov, V. E., "Continuous medium model for fractal media," Physics Letters A, Vol. 336, 2-3, 2005.

11. Muslih, S. and D. Baleanu, "Fractional multipoles in fractional space," Nonlinear Analysis: Real World Applications, Vol. 8, 198-203, 2007.
doi:10.1016/j.nonrwa.2005.07.001

12. Baleanu, D., A. K. Golmankhaneh, and A. K. Golmankhaneh, "On electromagnetic field in fractional space," Nonlinear Analysis: Real World Applications, Vol. 11, No. 1, 288-292, 2010.
doi:10.1016/j.nonrwa.2008.10.058

13. Wang, Z.-S. and B.-W. Lu, "The scattering of electromagnetic waves in fractal media," Waves in Random and Complex Media, Vol. 4, No. 1, 97-103, 1994.

14. Zubair, M., M. J. Mughal, and Q. A. Naqvi, "The wave equation and general plane wave solutions in fractional space," Progress In Electromagnetics Research Letters, Vol. 19, 137-146, 2010.

15. Oldham, K. B. and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.

16. Hussain, A. and Q. A. Naqvi, "Fractional rectangular impedance waveguide," Progress In Electromagnetics Research, Vol. 96, 101-116, 2009.
doi:10.2528/PIER09060801

17. Naqvi, Q. A., "Planar slab of chiral nihility metamaterial backed by fractional dual/PEMC interface," Progress In Electromagnetics Research, Vol. 85, 381-391, 2008.
doi:10.2528/PIER08081201

18. Naqvi, Q. A., "Fractional dual interface in chiral nihility medium," Progress In Electromagnetics Research Letters, Vol. Progress, 135-142, 2009.
doi:10.2528/PIERL09032405

19. Naqvi, Q. A., "Fractional dual solutions in grounded chiral nihility slab and their e®ect on outside fields," Journal of Electromagnetic Waves and Applications, Vol. 23, No. 5--6, 773-784, 2009.
doi:10.1163/156939309788019958

20. Naqvi, A., S. Ahmed, and Q. A. Naqvi, "Perfect electromagnetic conductor and fractional dual interface placed in a chiral nihility medium," Journal of Electromagnetic Waves and Applications, Vol. 24, No. 14--15, 1991-1999, 2010.

21. Naqvi, A., A. Hussain, and Q. A. Naqvi, "Waves in fractional dual planar waveguides containing chiral nihility metamaterial," Journal of Electromagnetic Waves and Applications, Vol. 24, No. 11--12, 1575-1586, 2010.
doi:10.1163/156939310792149614

22. Veliev, E. I., M. V. Ivakhnychenko, and T. M. Ahmedov, "Fractional boundary conditions in plane waves diffraction on a strip," Progress In Electromagnetics Research, Vol. 79, 443-462, 2008.
doi:10.2528/PIER07102406

23. Naqvi, S. A., M. Faryad, Q. A. Naqvi, and M. Abbas, "Fractional duality in homogeneous bi-isotropic medium," Progress In Electromagnetics Research, Vol. 78, 159-172, 2008.
doi:10.2528/PIER07090701

24. Muslih, S. I. and O. P. Agrawal, "A scaling method and its applications to problems in fractional dimensional space," J. Math. Physics, Vol. 50, No. 12, 123501-11, 2009.
doi:10.1063/1.3263940

25. Sangawa, U., H. T. Ewe, and S. L. Tan, "The origin of electromagnetic resonances in three-dimensional photonic fractals," Progress In Electromagnetics Research, Vol. 94, 153-173, 2009.
doi:10.2528/PIER09062203

26. Teng, H. T., H. T. Ewe, and S. L. Tan, "Multifractal dimension and its geometrical terrain properties for classification of Multiband multi-polarized SAR image," Progress In Electromagnetics Research, Vol. 104, 221-237, 2010.
doi:10.2528/PIER10022001

27. Mahatthanajatuphat, C., S. Saleekaw, P. Akkaraekthalin, and M. Krairiksh, "A rhombic patch monopole antenna with modified minkowski fractal geometry for UMTS, WLAN, and mobile WiMAX application," Progress In Electromagnetics Research, Vol. 89, 57-74, 2009.
doi:10.2528/PIER08111907

28. Mahatthanajatuphat, C., P. Akkaraekthalin, S. Saleekaw, and M. Krairiksh, "A bidirectional multiband antenna with modified fractal slot FED by CPW," Progress In Electromagnetics Research, Vol. 95, 59-72, 2009.
doi:10.2528/PIER09061603

29. Karim, M. N. A., M. K. A. Rahim, H. A. Majid, O. B. Ayop, M. Abu, and F. Zubir, "Log periodic fractal Koch antenna for UHF band applications," Progress In Electromagnetics Research, Vol. 100, 201-218, 2010.
doi:10.2528/PIER09110512

30. Siakavara, K., "Novel fractal antenna arrays for satellite networks: Circular ring sierpinski carpet arrays optimized by genetic algorithms ," Progress In Electromagnetics Research, Vol. 103, 115-138, 2010.
doi:10.2528/PIER10020110

31. He, Y., L. Li, C. H. Liang, and Q. H. Liu, "EBG structures with fractal topologies for ultra-wideband ground bounce noise suppression," Journal of Electromagnetic Waves and Applications, Vol. 24, No. 10, 1365-1374, 2010.
doi:10.1163/156939310791958734

32. Abramowitz, M. and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," U.S. Department of Commerce., 1972.

33. Balanis, C. A., Advanced Engineering Electromagnetics, Wiley, New York, 1989.

34. GÄotz, , M., , M. Rapp, and K. Dostert, "Power line channel characteristics and their e®ect on communication system design," IEEE Commun. Mag., Vol. 42, No. 4, 78-86, 2004.
doi:10.1109/MCOM.2004.1284933


© Copyright 2014 EMW Publishing. All Rights Reserved