Solving an initial value problem in inhomogeneous electrically and magnetically anisotropic uniaxial media

An initial value problem for a system of the second order partial differential equations, describing the electric wave propagation in vertically inhomogeneous electrically and magnetically anisotropic uniaxial media, is the main object of the study. The present paper suggests and justifies a new algorithm for solving this problem. This algorithm has several steps. On the first step the original initial value problem is written in terms of the Fourier images with respect to lateral variables. After that the obtained problem is transformed into an equivalent second kind vector integral equation of the Volterra type. A solution of this integral equation is constructed by successive approximations. At last, using the real Paley-Wiener theorem, a solution of the original initial value problem is found.