The paper deals with systems of linear partial differential equations of the form
where is an unknown vector-function and , are constant complex matrices. It is supposed that every eigenvalue of is positive ( denotes the conjugate transpose of ). Assuming initial and boundary conditions (2) , , the authors seek the solution of (1), (2) in the form of a Fourier series with respect to , whose coefficients depend on . An approximate solution is obtained by truncating this series and by suitable approximation of its coefficients. The estimate of the error is proved.