Accurate numerical scheme for solving fractional diffusion-wave two-step model for nanoscale heat conduction

Abstract

Recently, we have presented a fractional two-step model and its numerical method for nanoscale heat conduction. The model was obtained by introducing the Knudsen number (Kn) and the fractional-order (0<α<1) derivative in time to the parabolic two-step energy transport equations. For the case of 0<α<1, the model governs the ultraslow diffusion and we may call it the sub-diffusion two-step model. In this article, we extend this study to the case of 1<α<2 for that we may call the diffusion-wave two-step model, which can govern the intermediate processes for nanoscale heat conduction. The numerical scheme is developed based on the L1 approximation for fractional derivatives and the compact finite difference scheme for spatial derivative. Stability and convergence of the obtained numerical scheme are analyzed theoretically. We finally test the accuracy and applicability of the new model and its numerical scheme by three examples. By changing values of the Knudsen number and fractional-order derivative as well as the parameter in the boundary condition, the simulation could be a tool for analyzing the nanoscale heat conduction in intermediate processes such as porous and impure thin films exposed to ultrashort-pulsed lasers.