Mathematical problems of electromagnetic waves in layered media

In the first part of the thesis, the method of the approximate computation of the magnetic and electric matrix Green's functions in the rectangular parallelepiped with the perfect conducting boundary and the multi-layered isotropic structures has been suggested and investigated. This method based on the Fourier series expansion meta-approach. The method has been implemented in MATLAB and the error analysis of this method has been done by MATLAB tools. The computational experiments confirmed the robustness of the method. Moreover it has been shown that the method is applicable for the numerical computation of the electromagnetic wave propagation in rectangular parallelepiped arising from the pulse point sources. The modern computer tools allow us to observe the electromagnetic wave propagation. In the second part of the thesis the analytical method of solving the initial value problem for the time-dependent Maxwell's equations in the general electrically and magnetically anisotropic vertical inhomogeneous media has been described. This method is based on the following. The Maxwell's equations are written as symmetric hyperbolic system of the first order. This system is written in terms of the Fourier images with respect to lateral variables. The coefficients of this system are functions depending on vertical variable only. Applying method of integration along the characteristics of the symmetric hyperbolic system reduces to a system of Volterra integral equations. Method of successive approximations is applied for solving the obtained system of integral equations. Finally, the inverse Fourier transform is applied to find a solution of the initial value problem for the time-dependent Maxwell's equations. The methods and techniques of the generalized functions and partial differential equations of hyperbolic and elliptic types as well as the theory and application of the mathematical and functional analysis, and computational methods and computational tools are actively applied and used in the thesis.