A polynomial approach to determine the time-dependent electric and magnetic fields in anisotropic materials by symbolic computations Valery Yakhno Meltem Altunkaynak Article information:To cite this document: Valery Yakhno Meltem Altunkaynak , (2016),"A polynomial approach to determine the time-dependent electric and magnetic fields in anisotropic materials by symbolic computations", COMPEL: The

A new analytical method for solving an initial value problem (IVP) for the system of crystal optics with polynomial data and a polynomial inhomogeneous term is suggested. The found solution of the IVP is a polynomial. Theoretical and computational analysis of polynomial solutions and their comparison with non-polynomial solutions corresponding to smooth data are given. The applicability of polynomial solutions to physical processes is discussed. An implementation of this method has been made by symbolic computations in Maple 10.

A new method for the computation of a solution of the Maxwell's equations in bi‐anisotropic media with polynomial input is suggested. This method consists of the reduction of Maxwell's equations to a symmetric hyperbolic system of the first‐order partial differential equations. The solutions of the obtained system are constructed by symbolic computation in Maple for the given polynomial data in an explicit form. For the case when input data are given as smooth functions, an interpolation of the data has been made by polynomials and then an approximate solution has been computed by our method. The robustness of the method is confirmed by computational experiments. The comparison of the suggested method with finite difference time domain method has been done.

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In this paper, the three‐dimensional initial value problem for elastic system in inhomogeneous orthotropic media is considered and an analytical method is studied to solve this problem. The system is written in terms of Fourier images of displacements with respect to lateral variables. The resulting problem is reduced to integral equations of the Volterra type, whose solution is obtained by the method of successive approximations. Finally, using the real Paley‐Wiener theorem, it is shown that the solution of the initial value problem can be found by the inverse Fourier transform.