The present paper is devoted to the development of the method for searching an approximate solution of non-stationary problems on a variable interval. This method was first proposed by L.I. Slepyan. Here, the field of its applicability has been expanded to systems of equations, including equations of both hyperbolic and parabolic type. We describe the procedure of such approach in detail on a number of classical partial differential equations and compare the obtained results with exact analytical solution. Also, we consider a dynamic non-coupled thermoelastic problem. The method of expansion on a variable interval allows us to estimate the material response to a thermal disturbance at a large distance from the source.