volume | PIER Journals

Abstract

New fractional boundary conditions (FBC) on plane boundaries are introduced. FBC act as intermediate case between perfect electric conductor and perfect magnetic conductor. In certain sense FBC are analogue of commonly used impedance boundary conditions with pure imaginary impedance. The relation between fractional order and impedance is shown. Plane wave diffraction problem by a strip described by FBC is formulated and solved using new method which extends known methods. Numerical results for physical characteristics are presented. Analyzing the scattering properties of the fractional strip new features are observed. FBC has one important special case where the fractional order equals to 1/2. For this special case the solution of diffraction problem can be found in analytical form for any value of wavenumber. Also for small values of wavenumber monostatic radar cross section has new specific resonances which are absent for other values of fractional order.

1. Engheta, N., "Use of fractional integration to propose some 'Fractional' solutions for the scalar Helmholtz equation," Progress In Electromagnetics Research, Vol. 12, 107-132, 1996. Google Scholar

2. Engheta, N., "On the role of fractional calculus in electromagnetic theory," IEEE Antennas and Propagation Magazine, Vol. 39, No. 4, 35-46, 1997.
doi:10.1109/74.632994 Google Scholar

3. Engheta, N., "Fractional paradigm in electromagnetic theory," Frontiers in Electromagnetics, 2000. Google Scholar

4. Samko, S. G., A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theoryand Applications, 1993.

5. Engheta, N., "Fractional curl operator in electromagnetics," Microwave and Optical TechnologyL etters, Vol. 17, No. 2, 86-91, 1998.
doi:10.1002/(SICI)1098-2760(19980205)17:2<86::AID-MOP4>3.0.CO;2-E Google Scholar

6. Naqvi, Q. A. and A. A. Rizvi, "Fractional dual solutions and corresponding sources," Progress In Electromagnetics Research, Vol. 25, 223-238, 2000.
doi:10.2528/PIER99051801 Google Scholar

7. Veliev, E. I. and M. V. Ivakhnychenko, "Fractional curl operator in radiation problems," Proceedings of MMET*04, 231-233, 2004.

8. Veliev, E. I. and M. V. Ivakhnychenko, "Elementary fractional dipoles," Proceedings of MMET*06, 485-487, 2006.

9. Hussain, A. and Q. A. Naqvi, "Fractional curl operator in chiral medium and fractional non-symmetric transmission line," Progress In Electromagnetic Research, Vol. 59, 199-213, 2006.
doi:10.2528/PIER05092801 Google Scholar

10. Naqvi, Q. A., G. Murtaza, and A. A. Rizvi, "Fractional dual solutions to Maxwell equations in homogeneous chiral medium," Optics Communications, Vol. 178, 27-30, 2000.
doi:10.1016/S0030-4018(00)00651-9 Google Scholar

11. Lakhtakia, A., "A representation theorem involving fractional derivatives for linear homogeneous chiral media," Microwave Opt. Tech. Lett., Vol. 28, 385-386, 2001.
doi:10.1002/1098-2760(20010320)28:6<385::AID-MOP1048>3.0.CO;2-L Google Scholar

12. Hussain, A., S. Ishfaq, and Q. A. Naqvi, "Fractional curl operator and fractional waveguides," Progress In Electromagnetics Research, Vol. 63, 319-335, 2006.
doi:10.2528/PIER06060604 Google Scholar

13. Veliev, E. and N. Engheta, "Fractional curl operator in reflection problems," Proceedings of MMET*04, 228-230, 2004.

14. Veliev, E. I.T. M. Ahmedov, and M. V. Ivakhnychenko, "New generalized electromagnetic boundaries — Fractional operators approach," Proceedings of MMET*06, 434-437, 2006.

15. Onufrienko, V. M., "Interaction of a plane electromagnetic wave with a metallized fractal surface," Telecommunications and Radio Engineering, Vol. 55, No. 3, 2001. Google Scholar

16. Engheta, N., "Fractionalization methods and their applications to radiation and scattering problems," Proceedings of MMET*00, Vol. 1, 34-40, 2000.

17. Ivakhnychenko, M. V., E. I. Veliev, and T. V. Ahmedov, "Fractional operators approach in electromagnetic wave reflection problems," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 13, 1787-1802, 2007. Google Scholar

18. Senior, T. B. and J. L. Volakis, Approximate BoundaryConditions in Electromagnetics, The institution of Electrical Engineers, 1995.

19. Honl, H., A. W. Maue, and K. Westpfahl, Theorie der Beugung, Springer-Verlag, 1961.

20. Veliev, E. and N. Engheta, "Generalization of Green's theorem with fractional differintegration," 2003 IEEE AP-S International Symposium & USNC/URSI National Radio Science Meeting, 2003.

21. Veliev, E. and T. M. Ahmedov, "Fractional solution of Helmholtz equation — A new presentation," Reports of NAS of Azerbaijan, No. 4, 20-27, 2005. Google Scholar

22. Uflyand, Y. S., "The method of dual equations in problems of mathematical physics," Nauka, 1977. Google Scholar

23. Veliev, E. and V. V. Veremey, "Numerical-analytical approach for the solution to the wave scattering by polygonal cylinders and flat strip structures," Analytical and Numerical Methods in Electromagnetic Wave Theory, 1993. Google Scholar

24. Veliev, E. and V. P. Shestopalov, "A general method of solving dual integral equations," Sov. Physics Dokl., Vol. 33, No. 6, 411-413, 1988. Google Scholar

25. Bateman, H. and A. Erdelyi, Higher Transcendental Functions, Vol. 2, 1953-1955, Vol. 2, 1953.

26. Prudnikov, H. P., Y. H. Brychkov, and O. I. Marichev, Special Functions, Vol. 2, Integrals and Series, 1986.

27. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, New York, 1972.

28. Herman, M. I. and J. L. Volakis, "High frequency scattering by a resistive strip and extensions to conductive and impedance strips," Radio Science, Vol. 22, No. 3, 335-349, 1987.
doi:10.1029/RS022i003p00335 Google Scholar

29. Ikiz, T., S. Koshikawa, K. Kobayashi, E. I. Veliev, and A. H. Serbest, "Solution of the plane wave diffraction problem by an impedance strip using a numerical-analytical method: E-polarized case," Journal of Electromagnetic Waves and Applications, Vol. 15, No. 3, 315-340, 2001.
doi:10.1163/156939301X00481 Google Scholar

30. Balanis, C. A., Advanced Engineering Electromagnetic, Wiley, 1989.

Dr. Eldar Veliev

Dr. Maxim Ivakhnychenko

Dr. Turab Ahmedov