Abstract: Multiple boundary conditions exist in different areas of the same boundary surface are called mixed boundary, which is a well-known mechanical problem. It is necessary to solve the mixed boundary value problem, when to accurately analyze such problems. For the general 3D non-axisymmetric situation, there are often mathematical difficulties in solving the mixed boundary value problem. In this paper, an analytical method for solving three-dimensional non-axisymmetric mixed boundary value problems is presented by using Hilbert's theorem and double Fourier transform. Basted on this method, the coupled vibration problem for a rectangular plate resting on a saturated porous half-space with mixed permeable boundary is studied. Firstly, the dynamic governing equations of rectangular plates and saturated porous foundations are established based on Kirchhoff theory and Biot's porous medium theory. The operator equation is decoupled by double Fourier transforms to obtain the general solution of the rectangular plate and the foundation. The mixed boundary value problem is converted to two pairs of two-dimensional dual integral equations, with the contact surface stress and pore pressure as the basic unknowns, and to be solved by the Schmidt's method. The displacement and internal force analytical equations of the coupled vibration of the plate-foundation system are obtained. Finally, numerical examples are given to analyze and discuss the vibration response and parameters of the rectangular substrate on the saturated half space.