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.