V.G. Yakhno, T.M. Yakhno, M. Kasap
Dokuz Eylul University
35160, Izmir, Turkey
email: yakhno@cs.deu.edu.tr
To understand how electromagnetic waves propagate in anisotropic media is a very important question when we study the properties of new materials [1]. Mathematical model of electromagnetic wave propagation in an anisotropic 3-D space is described by Maxwell's system with an arbitrary positive definite symmetric matrix of dielectric permeability and the magnetic permeability as a positive constant. The electromagnetic waves are exited by a pulse external current concentrated at a fixed point. The explicit formulas for generalized solutions of initial value problems for this Maxwell system with different types of anisotropy are obtained. These formulas have the form of a linear combination of the Dirac delta functions with the support on the fronts of three waves and its derivatives, Heaviside step function, with the support inside the wave fronts. These explicit formulas were described in details in [2]. >From one hand these formulas look rather cumbersome and complicated but from the other hand they have quite regular structure and symmetry. This allows us to make a decomposition of the formulas and calculate in parallel the values of points for different fragments of the images. This parallelization essentially reduces the runtime of the program. The advantage of using explicit formulas consists in a clear way for modeling and simulation of the waves propagation in anisotropic media. We used MatLAB to generate 3D images and animated movies of wave propagation. These images are collected in the library of images available in [3]. We note that the simulation of the electromagnetic fields based on explicit formulas is the best one because it gives us exact images. But unfortunately it is impossible to find explicit formulas in the general case and we need to use approximate solutions generated by numerical methods. Our library can serve as a collection of patterns and samples when we analyze the structure of anisotropic materials or evaluate the performance of numerical methods.
References:
1. Cohen G.C, Heikkola E., Joly P. Neittaan Maki P. Mathematical and Numerical
Aspects of Waves Propagation, Springer Verlag, Berlin, 2003.
2.Yakhno, V.G., Yakhno, T.M., Kasap M. Simulation of
Electromagnetic Wave Propagation in Anisotropic Media. Selcuk Journal of
Applied Mathematics, V. 4, N. 2, p.113122, 2003.
3. http://www.cs.deu.edu.tr/mustafakasap/album