volume | PIER Journals

Abstract

A two-dimensional Green's function for a half space geometry, comprising planar interface only due to two different non-integer dimensional spaces, has been derived. Medium hosting the time harmonic electric line source and planar interface is homogeneous and isotropic. Radiated field is written in terms of unknown spectrum of plane waves. Unknown spectrum functions are determined using the related boundary conditions. It has been shown that although wavenumbers of both half spaces are same, due to difference of dimensions of the two half spaces, reflection and transmission occur. When dimensions of both half spaces are taken equal to two, derived expressions yield field radiated by a line source in an unbounded homogeneous medium with integer dimensional space.

1. Stillinger, F. H., "Axiomatic basis for spaces with non-integer dimensions," Journal of Mathematical Physics, Vol. 18, 1224-1234, 1977.
doi:10.1063/1.523395 Google Scholar

2. Zubair, M., M. J. Mughal, and Q. A. Naqvi, Fractional Fields and Waves in Fractional Dimensional Space, Springer-Verlag, 2012.
doi:10.1007/978-3-642-25358-4

3. Zubair, M., M. J. Mughal, and Q. A. Naqvi, "The wave equation and general plane wave solution in fractional space," Progress In Electromagnetics Research Letters, Vol. 19, 137-146, 2010.
doi:10.2528/PIERL10102103 Google Scholar

4. Zubair, M., M. J. Mughal, and Q. A. Naqvi, "An exact solution of cylindrical wave equation for electromagnetic field in fractional dimensional space," Progress In Electromagnetics Research, Vol. 114, 443-455, 2011.
doi:10.2528/PIER11021508 Google Scholar

5. Zubair, M., M. J. Mughal, and Q. A. Naqvi, "An exact solution of spherical wave in D-dimensional fractional space," Journal of Electromagnetic Waves and Applications, Vol. 25, No. 10, 1481-1491, 2011. Google Scholar

6. Zubair, M., M. J. Mughal, and Q. A. Naqvi, "On electromagnetic wave propagation in fractional space," Nonlinear Analysis: Real World Applications, Vol. 12, 2844-2850, 2011.
doi:10.1016/j.nonrwa.2011.04.010 Google Scholar

7. Zubair, M., 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 Google Scholar

8. Asad, H., M. Zubair, and M. J. Mughal, "Reflection and transmission at dielectric-fractal interface," Progress In Electromagnetics Research, Vol. 125, 543-558, 2012.
doi:10.2528/PIER12012402 Google Scholar

9. Attiya, A. M., "Reflection and transmission of electromagnetic wave due to a quasi-fractional-space slab," Progress In Electromagnetics Research Letters, Vol. 24, 119-128, 2011.
doi:10.2528/PIERL11051105 Google Scholar

10. Baleanua, D., A. K. Golmankhaneh, and A. K. Golmankhaneh, "On electromagnetic field in fractional space," Nonlinear Analysis: Real World Applications, Vol. 11, 288-292, 2010.
doi:10.1016/j.nonrwa.2008.10.058 Google Scholar

11. Naqvi, Q. A. and M. Zubair, "On cylindrical model of electrostatic potential in fractional dimensional space," Optik-International Journal for Light and Electron Optics, Vol. 127, 3243-3247, 2016.
doi:10.1016/j.ijleo.2015.12.019 Google Scholar

12. Noor, A., A. A. Syed, and Q. A. Naqvi, "Quasi-static analysis of scattering from a layered plasmonic sphere in fractional space," Journal of Electromagnetic Waves and Applications, Vol. 29, No. 8, 1047-1059, 2015.
doi:10.1080/09205071.2015.1032436 Google Scholar

13. Asad, H., M. J. Mughal, M. Zubair, and Q. A. Naqvi, "Electromagnetic Green’s function for fractional space," Journal of Electromagnetic Waves and Applications, Vol. 26, No. 14-15, 1903-1910, 2012.
doi:10.1080/09205071.2012.720748 Google Scholar

14. Abbas, M., A. A. Rizvi, and Q. A. Naqvi, "Two dimensional Green’s function for non-integer dimensional dielectric half space geometry," Optik-International Journal for Light and Electron Optics, Vol. 127, 8530-8535, 2016.
doi:10.1016/j.ijleo.2016.06.059 Google Scholar

15. Zubair, M. and L. K. Ang, "Fractional-dimensional Child-Langmuir law for a rough cathode," Physics of Plasmas (1994-present), Vol. 23, 072118, 2016.
doi:10.1063/1.4958944 Google Scholar

16. Hameed, A., M. Omar, A. A. Syed, and Q. A. Naqvi, "Power tunneling and rejection from fractal chiral-chiral interface," Journal of Electromagnetic Waves and Applications, Vol. 28, No. 14, 1766-1776, 2014.
doi:10.1080/09205071.2014.938448 Google Scholar

17. Abbas, M., A. A. Rizvi, A. Fiaz, and Q. A. Naqvi, "Scattering of electromagnetic plane wave from a low contrast circular cylinder buried in non-integer dimensional half space," Journal of Electromagnetic Waves and Applications, Vol. 31, No. 3, 263-283, 2017.
doi:10.1080/09205071.2016.1276859 Google Scholar

18. Sandev, T., I. Petreska, and E. K. Lenzi, "Harmonic and anharmonic quantum-mechanical oscillators in non-integer dimensions," Phys. Lett. A, Vol. 378, 109-116, 2014.
doi:10.1016/j.physleta.2013.10.048 Google Scholar

19. Palmer, C. and P. N. Stavrinou, "Equations of motion in a non-integer-dimensional space," Journal of Physics A: Mathematical and General, Vol. 37, 2004. Google Scholar

20. Muslih, S., D. Baleanu, and E. Rabei, "Gravitational potential in fractional space," Open Physics, Vol. 5, 285-292, 2007.
doi:10.2478/s11534-007-0014-9 Google Scholar

21. Tarasov, V. E., "Continuous medium model for fractal media," Phys. Lett. A, Vol. 336, 167-174, 2005.
doi:10.1016/j.physleta.2005.01.024 Google Scholar

22. Tarasov, V. E., "Anisotropic fractal media by vector calculus in non-integer dimensional space," Journal of Mathematical Physics, Vol. 55, 083510, 2014.
doi:10.1063/1.4892155 Google Scholar

23. Tarasov, V. E., "Elasticity of fractal materials using the continuum model with non-integer dimensional space," Comptes Rendus Mecanique, Vol. 343, 57-73, 2015.
doi:10.1016/j.crme.2014.09.006 Google Scholar

24. Tarasov, V. E., "Flow of fractal fluid in pipes: Non-integer dimensional space approach," Chaos, Solitons and Fractals, Vol. 67, 26-37, 2014.
doi:10.1016/j.chaos.2014.06.008 Google Scholar

25. Tarasov, V. E., "Vector calculus in non-integer dimensional space and its applications to fractal media," Communications in Nonlinear Science and Numerical Simulation, Vol. 20, 360-374, 2015.
doi:10.1016/j.cnsns.2014.05.025 Google Scholar

26. Tarasov, V. E., "Electromagnetic waves in non-integer dimensional spaces and fractals," Chaos, Solitons and Fractals, Vol. 81, 38-42, 2015.
doi:10.1016/j.chaos.2015.08.017 Google Scholar

27. Tarasov, V. E., "Fractal electrodynamics via non-integer dimensional space approach," Phys. Lett. A, Vol. 379, 2055-2061, 2015.
doi:10.1016/j.physleta.2015.06.032 Google Scholar

28. Tarasov, V. E., "Acoustic waves in fractal media: Non-integer dimensional spaces approach," Wave Motion, Vol. 63, 18-22, 2016.
doi:10.1016/j.wavemoti.2016.01.003 Google Scholar

29. Naqvi, Q. A. and A. A. Rizvi, "Scattering from a cylindrical object buried in a geometry with parallel interfaces," Progress In Electromagnetics Research, Vol. 27, 19-35, 2000.
doi:10.2528/PIER98120902 Google Scholar

30. Bender, C. M. and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineering, McGraw-Hill, 1999.
doi:10.1007/978-1-4757-3069-2

Dr. Muhammad Fiaz

Dr. Qaisar Abbas Naqvi