Abstract <p>A linear system of classical and hyperbolic thermoelasticity has been established in the framework of the Caputo time fractional derivative in the cartesian domain. The solutions of the homogeneous time fractional system of classical and hyperbolic thermoelasticity with respect to initial conditions are obtained by applying the fractional natural decomposition method (FNDM). The convergence of infinite series solutions has been addressed. The stability conditions of the proposed systems are discussed. Furthermore, the physical behavior of the acquired solutions has been represented in the form of graphical representations for different fractional orders. The obtained results of the study demonstrate the FNDM’s high accuracy and computational effectiveness. Moreover, the significant role of relaxation time and the fractional order parameters are studied as material characteristics.</p>