A new approach to the solution of initial boundary value problems is proposed. It is based on defining integral relations connecting right sides of different types of boundary conditions. It is assumed that one of these solutions has been found. Right sides of boundary conditions of the other problem, being integral equation solutions, are defined through quadrature formulae. Then, solution of this problem assumes as Green's function convolution of the first problem with obtained solutions of integral equations. Non-stationary problem of elastic diffusion for half-space is used as an example.