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

Anomalous diffusion phenomena are widely observed in biological tissues due to microstructural heterogeneity and nonlocal transport mechanisms. Time--space fractional diffusion equations provide a powerful mathematical framework to capture such effects; however, their numerical approximation remains challenging because of nonlocality, memory requirements, and stability constraints. In this work, we propose an efficient and energy-structure-preserving numerical method for a class of time--space fractional diffusion equations modelling anomalous transport in heterogeneous biological tissues. The method combines a memory-efficient temporal discretisation of the fractional derivative with a stable spatial approximation of the fractional diffusion operator. Rigorous stability and convergence analyses are established under mild regularity assumptions. Numerical experiments confirm the theoretical error estimates and illustrate the impact of tissue heterogeneity and fractional parameters on diffusion behaviour. The proposed scheme preserves the coercive energy structure of the continuous problem and ensures unconditional stability at the discrete level.