The dynamic equations for quasicrystals are written as time-dependent partial
differential equations of the second order relative to phonon and phason
displacements. In these equations phonons describe the dynamics of wave
propagation and phasons describe diffusion process in quasicrystals. A new
approach for deriving a solution (phonon and phason displacements) of the
initial value problem is proposed. In this approach the Fourier transform with
respect to 3D space variable of the given phonon, phason forces and initial
displacements are assumed to be vector functions with components which have finite
supports with respect to Fourier parameters for every fixed time variable. The
equations for the Fourier images of displacements are reduced to a vector
integral equation of the Volterra-type depending on Fourier parameters.
The solution of the obtained vector integral equation is solved by successive
approximations. Finally, phonon and phason displacements are derived by matrix
transformations and the inverse Fourier transform to the solution of the vector
integral equation.
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