Computational analysis of fractional heat conduction with fading memory

Abstract

This study presents a computational analysis of a fractional-order model for heat conduction in complex media with fading memory. The model incorporates Caputo time-fractional derivatives of order α∈(0,1), accounts for heat flux memory effects, and includes a neutral delay. By representing the relaxation functions of heat flux and heat capacity as finite linear combinations of decaying exponentials, we derive a coupled system involving both fractional temporal operators and classical time derivatives, which extends the original fractional heat-conduction equation with two auxiliary equations. The stability estimate for the solution of the resulting system is established in a finite-dimensional Hilbert space, with respect to initial conditions and source terms. For the computational implementation, we first propose a difference scheme based on the L1 formula and rigorously investigate its unconditional stability, demonstrating a temporal convergence rate of order min{2-α,1+α}. To achieve higher accuracy that is independent of the fractional order, an additional scheme based on the L2 formula is developed and proven to exhibit second-order temporal convergence. In addition, the methods are extended to graded non-uniform meshes to enhance their accuracy in cases where the solution possesses limited initial smoothness. Numerical simulations are conducted to validate the theoretical results.