Summary: Hyperbolic linear theory of heat propagation has been established in the framework of a Caputo time fractional order derivative. The solution of a system of integer and fractional order initial value problems is achieved by employing the Adomian decomposition approach. The obtained solution is in convergent infinite series form, demonstrating the method’s strengths in solving fractional differential equations. Moreover, the double Laplace transform method is employed to acquire the solution of a system of integer and fractional order boundary conditions in the Laplace domain. An inversion of double Laplace transforms has been achieved numerically by employing the Xiao algorithm in the space-time domain. Considering the non-Fourier effect of heat conduction, the finite speed of thermal wave propagation has been attained. The role of the fractional order parameter has been examined scientifically. The results obtained by considering the fractional order theory and the integer order theory perfectly coincide as a limiting case of fractional order parameter approaches one.