Numerical Method for Solving the Time-Fractional Dual-Phase-Lagging Heat Conduction Equation with the Temperature-Jump Boundary Condition

Abstract

This article proposes a new nanoscale heat transfer model based on the Caputo type fractional dual-phase-lagging (DPL) heat conduction equation with the temperature-jump boundary condition. The model is proved to be well-posed. A finite difference scheme based on the L1 approximation for the Caputo derivative is then presented for solving the fractional DPL model. Unconditional stability and convergence of the scheme are proved by using the discrete energy method. Three numerical examples are given to verify the accuracy of the scheme. Results show the convergence order to be \(O(\tau ^{2-\alpha }+h^2)\) , which coincides with the theoretical analysis. A simple nanoscale semiconductor silicon device is illustrated to show the applicability of the model. It is seen from the numerical result that when \(\alpha =1\), the fractional DPL reduces to the conventional DPL and the obtained peak temperature is almost identical to those obtained in the literature. However, when \(\alpha <1\), the model predicts a higher peak temperature level than that when \(\alpha =1\). In particular, when \(\alpha = 0.7\) and 0.9, an oscillatory temperature at the beginning is observed. This indicates that the fractional DPL model can be an excellent candidate for analyzing the temperature instability appearing in electronic nano-semiconductor devices.