Numerical inversion of the fractional derivative index and surface thermal flux for an anomalous heat conduction model in a multi-layer medium

In this paper, we study an inverse problem for identifying multiple parameters of an anomalous heat conduction model with fractional derivative flux boundary conditions in a composite medium. The anomalous heat conduction equation in a layered medium is derived using the conservation of energy and the modified Fourier law with a Riemann-Liouville fractional operator. For the forward problem, we construct an effective finite difference scheme by applying the balance method to deal with the discontinuity interface. For the inverse problem, we utilize the Levenberg-Marquardt (L-M) regularization technique combined with the Armijo rule to simultaneously identify the order of the fractional derivative and the left boundary heat flux term. Finally, we use experimental data extracted from the measurement in a carbon-carbon composite material to verify the effectiveness of the proposed technique. In the experiments, we analyze the sensitivity coefficients and show the numerical results graphically. Furthermore, we introduce a diagonal scaling matrix transformation to alleviate the effect of the different parameter magnitudes on the inverse problem. The simulation results confirm that the fractional heat conduction model with estimated parameters gives a more accurate and adequate description for the heat transfer process in the carbon-carbon composite material.