A Semi-Explicit Algorithm for Parameters Estimation in a Time-Fractional Dual-Phase-Lag Heat Conduction Model

Abstract

This paper presents a new semi-explicit algorithm for parameters estimation in a time-fractional generalization of a dual-phase-lag heat conduction model with Caputo fractional derivatives. It is shown that this model can be derived from a general linear constitutive relation for the heat transfer by conduction when the heat conduction relaxation kernel contains the Mittag–Leffler function. The model can be used to describe heat conduction phenomena in a material with power-law memory. The proposed algorithm of parameters estimation is based on the time integral characteristics method. The explicit representations of the thermal diffusivity and the fractional analogues of the thermal relaxation time and the thermal retardation are obtained via a Laplace transform of the temperature field and utilized in the algorithm. An implicit relation is derived for the order of fractional differentiation. In the algorithm, this relation is resolved numerically. An example illustrates the proposed technique.