Bio-heat transfer analysis based on fractional derivative and memory-dependent derivative heat conduction models

The fractional derivative (FD) and memory-dependent derivative (MDD) are efficient methods to predict the transient thermal responses. In the present paper, for the first time, the FD and MDD are introduced into classical Pennes bio-heat conduction equation with single phase-lag and the corresponding bio-heat transfer models have been formulated with the aids of thermal energy conservation equation. The Laplace transform and numerical inverse transform method are adopted to get access to thermal responses due to temperature shock. Reductions of the present models to classical Pennes and single phase-lag bio-heat transfer models are presented and comparison between different FD and MDD in transient bio-heat transfer is carried out firstly. An approximate formulation of propagation speed of thermal signal in FD models is given according to results firstly. Influences of fractional derivative parameter and tempered parameter in FD, delay time in MDD, phase-lag time and rate of blood perfusion on temperature variation and distribution are carried out numerically. Both fractional derivative and memory-dependent derivative have the ability to predict thermal response between diffusion and wave behavior and CF fractional derivative and MDD with K3 may be suitable methods.