It is well known that the heat conduction in the Pennes bioheat transfer equation is based on Fourier law
. To incorporate the non-Fourier behavior, Cattaneo and Vernott proposed a modified heat conduction equation as follows [10, 11]:
In the case where the two phase-lags are exactly the same, [[tau].sub.q] = [[tau].sub.T], there would bean instantaneous response between the temperature gradient and the heat flux, and (1) would be equivalent to the classical Fourier law
, except for a shift in time.
The generalised thermoelastic theory, which is based on a new heat conductive law to replace the classical Fourier law
, has been successfully used to explore the thermal shock problem.
Illustration of the first Fourier law
: A) Heat conduction between two parallel plates; B) The temperature profile inside the wall
In some cases, the Fourier law
cannot explain, such as near the absolute zero temperature, thermal gradient, which is extreme .
The local fractional transient heat conduction equations based upon the Fourier law within local fractional derivative arising in heat transfer from discontinuous media were presented in [21-24].
Fourier law of heat conduction in fractal medium with local fractional derivative is expressed by 
Tzou [11, 12] had introduced another modification to Fourier law
, by inventing two time lags, Dual Phase Lag (DPL), between the heat flux and the temperature gradient namely the heat flux time lag and the temperature gradient time lag.
This leads for isotropic systems to the Fourier law
More precisely, Fourier law
is diffusive and cannot predict the finite temperature propagation speed in transient situations, in this context, the Cattaneo-Vernotte equation corrects the nonphysical property of infinite propagation of the Fourier and Fickian theory of the diffusion of heat, and this equation also known as the telegraph equation for the temperature is a generalization of the heat diffusion (Fourier's law) and particle diffusion (Fick's laws) equations.
When both lags are zero, the Fourier law
is recovered, while for [[tau].sub.q] > 0 and [[tau].sub.T] = 0, it reduces to the Single-Phase-Lag (SPL) model .
The time-nonlocal generalization of the Fourier law
with the "long-tail" power kernel [11, 13-15] can be interpreted in terms of fractional calculus (theory of integrals and derivatives of noninteger order) and results in the time-fractional heat conduction equation