The present work deals with a new problem of generalized thermoelasticity with one relaxation time for an infinitely long and isotropic annular cylinder of temperature dependent physical properties. The inner and outer curved surfaces of the cylinder are subjected to both the mechanical and thermal boundary conditions. A finite difference model is developed to derive the solution of the problem in which the governing equations are coupled non linear partial differential equations. The transient solution at any time can be evaluated directly from the model. In order to demonstrate the efficiency of the present model we consider a suitable material and obtain the numerical solution of displacement, temperature, and stresses inside the annulus for both the temperature-dependent and temperature-independent material properties of the medium. The results are analyzed with the help of different graphical plots.