Numerical simulation of anomalous diffusion with robin boundary conditions: a high-accuracy spectral collocation approach

Abstract

Abstract A novel spectral collocation method is developed for the time-fractional diffusion equation (TFDE) with an initial condition (IC) and two homogeneous Robin boundary conditions (HRBCs), modeling anomalous transport in complex media. We utilize generalized shifted Jacobi polynomials (GSJPs) to generate basis functions that satisfy the boundary conditions by design. The problem is reduced to a simple algebraic system via operational matrices (OMs). We provide a complete error analysis and demonstrate the method’s effectiveness on two test problems: (i) non-homogeneous heat transfer in fractal media, and (ii) autocatalytic reaction-diffusion dynamics. For these cases, our approximate solutions (APPSs) are compared against exact solutions and existing numerical results. The simulations confirm that the proposed scheme captures the subtle memory effects and steep gradients of generation-diffusion processes with high spectral accuracy (10 −14 ), significantly outperforming established L1-Galerkin benchmarks.