A Fractional-Order Alternative for Phase-Lagging Equation

Abstract

Phase-lagging equation (PLE) is an equation describing micro/nano scale heat conduction, where the lagging response must be included, particularly under low temperature or high heat-flux conditions. However, finding the analytical or numerical solutions of the PLE is tedious in general. This article aims at seeking a fractional-order heat equation that is a good alternative for the PLE. To this end, we consider the PLE with simple initial and boundary conditions and obtain a fractional-order heat equation and an associated numerical method for approximating the solution of the PLE. In order to better approximate the PLE, the Levenberg-Marquardt iterative method is employed to estimate the optimal parameters in the fractional-order heat equation. This fractional-order alternative is then tested and compared with the PLE. Results show that the fractional method is promising.