Fractional thermal diffusion and the heat equation

Fractional calculus is the branch of mathematical analysis that deals with operators interpreted as derivatives and integrals of non-integer order. This mathematical representation is used in the description of non-local behaviors and anomalous complex processes. Fourier's law for the conduction of heat exhibit anomalous behaviors when the order of the derivative is considered as 0 < beta, gamma <= 1 for the space-time domain respectively. In this paper we proposed an alternative representation of the fractional Fourier's law equation, three cases are presented; with fractional spatial derivative, fractional temporal derivative and fractional space-time derivative (both derivatives in simultaneous form). In this analysis we introduce fractional dimensional parameters sigma(x) and sigma(t) with dimensions of meters and seconds respectively. The fractional derivative of Caputo type is considered and the analytical solutions are given in terms of the Mittag-Leffler function. The generalization of the equations in space-time exhibit different cases of anomalous behavior and Non-Fourier heat conduction processes. An illustrative example is presented.