An inverse problem to estimate relaxation parameter and order of fractionality in fractional single-phase-lag heat equation

In this paper, an inverse analysis is performed for simultaneous estimation of relaxation time and order of fractionality in fractional single-phase-lag heat equation. This fractional heat conduction equation is applied on two physical problems. In inverse procedure, solutions of a previously validated linear dual-phase-lag model on the physical problems under study have been used as the measured temperatures. The inverse fractional single-phase-lag heat conduction problem is solved using the nonlinear parameter estimation technique based on the Levenberg–Marquardt method. The results of the present study show that the Levenberg–Marquardt method can be successfully applied on the inverse fractional heat transfer problem. The solution procedures employed in the present study for direct and inverse problems have greatly increased the reliability and success of parameter estimation problem. In the present study, for the first time, relaxation time and fractionality of a non-homogeneous medium (i.e. processed meat) have been determined. Also, the results of this study show that the fractional single-phase-lag model can predict the same temperature distribution as the linear dual-phase-lag model for the problem under study. This latter result enables us to consider further generalization of the dual-phase-lag model to fractional dual-phase-lag models.