Numerical simulation and parameters estimation of the time fractional dual-phase-lag heat conduction in femtosecond laser heating

In this paper, considering the heat flux and the temperature gradient approximated by the fractional Taylor's series expansions with different orders in dual-phase-lag (DPL) model, we propose the time fractional DPL heat conduction model for thin gold films heating by femtosecond laser pulses. The solutions of the model are investigated analytically and numerically. Firstly, the semi-analytical solution is presented by the Laplace transform method and the Fourier cosine transform method. With the L1 approximation of the Caputo fractional derivative, the finite difference scheme is derived to describe the effects of femtosecond laser pulses on the temperature distributions of the gold films. The effectiveness of the numerical algorithm is examined by the comparison with the semi-analytical solution. And then based on the numerical algorithm and the experimental datasets of femtosecond laser pulses heating on gold films of various thicknesses, the estimations of the model parameters are considered by the modified hybrid Nelder–Mead simplex search and particle swarm optimization (MH-NMSS-PSO). Using the optimal values of the parameters estimation, we verify the consistencies between the semi-analytical solutions of the model and the datasets. It proves the accuracy of the time fractional DPL heat conduction model in characterizing the temperature distributions of thin gold films. Finally, the varieties of temperature distribution with time and position are considered and then the effects of the model parameters on the temperature distribution are also discussed.