Generalized Laplace Transform Based on the Differentiation Operator With Piecewise Constant Coefficients

The paper develops the theory of matrix integral Fourier transforms based on a differential operator with piecewise constant matrix coefficients. The definition of the matrix Fourier transform is given, its properties and applications to the modeling of interrelated wave processes in piecewise homogeneous media are studied. An inversion formula for the matrix integral Fourier transform is proved. Significant differences from the scalar case are revealed. A technique for applying the matrix Fourier transform to solving interrelated mixed boundary value problems for systems of hyperbolic differential equations with matrix piecewise constant coefficients is developed. A solution is found for the vector analog of the problem of wave propagation in an infinite string with two regions of different density. A vector analogue of the d'Alembert formula is found. A solution is obtained for a mixed initial-boundary value problem for a system of differential equations of parabolic type, which describes an $n$ component model of an interconnected process of heat and mass transfer in a two-layer media.