Derivation of a solution of dynamic equations of motion for quasicrystals

Abstract

The dynamic equations of motion in quasicrystals are written in terms of time-dependent partial differential equations of the second order relative to phonon and phason displacements. A method of derivation of a solution (phonon and phason displacements) of the initial value problem is proposed in this paper. In this method, images of the Fourier transform with respect to the 3D space variable of the given phonon, phason forces, and initial displacements are assumed to be vector functions with components which have finite supports for every fixed time variable. The Fourier images of displacements are computed by matrix transformations and solving ordinary differential equations, coefficients and non-homogeneous terms as well as initial data of which depend on 3D Fourier parameter. Finally, phonon and phason displacements are computed by the inverse Fourier transform to obtained Fourier image.