Efficient and Structure-Preserving Numerical Methods for Time–Space Fractional Diffusion in Heterogeneous Biological Tissues

Time–space fractional diffusion equations are widely used to model anomalous transport in heterogeneous biological tissues, where memory effects, spatial nonlocality, and coefficient variability are intrinsically coupled. However, existing numerical approaches typically treat these aspects in isolation, and a fully discrete framework that simultaneously accounts for heterogeneity, long-memory effects, and computational efficiency remains lacking. In this work, a fully discrete numerical method is developed and analyzed. The method integrates heterogeneous diffusion coefficients and memory-efficient temporal discretization within a unified variational framework. It combines a finite element approximation of a spectral fractional elliptic operator with an implicit L1 discretization of the Caputo derivative enhanced by a sum-of-exponentials approximation of the memory kernel. Unconditional stability, preservation of a discrete energy structure, and a fully discrete error estimate are established, explicitly separating temporal, spatial, and kernel approximation errors. The proposed approach reduces memory complexity from O(N) to O(logN) without compromising accuracy. Numerical experiments confirm the theoretical convergence rates, demonstrate stable behavior across all tested configurations, and illustrate the impact of heterogeneous coefficients on anomalous transport dynamics.