Distributed-order fractional diffusion equation with Hilfer–Prabhakar fractional derivative

Fractional diffusion equations (FDEs) are widely used to model anomalous diffusion in a variety of complex systems. In this work, we investigate a distributed-order FDE that incorporates the Hilfer–Prabhakar (HP) time-fractional derivative, which introduces a generalized memory kernel characterized by composite power-law scaling. This formulation extends the classical Riemann–Liouville (RL) and Caputo derivatives. We identify an admissible parameter regime under which the solution of the FDE behaves as a probability density function (PDF), and the HP derivative reduces to the Caputo-type under integer-order initial conditions. For the single-order case, we show that the PDF defined on an infinite domain can be expressed in terms of Fox H-functions in both short- and long-time limits, revealing distinctive subdiffusive behavior. Additionally, we derive asymptotic expressions for the mean squared displacement (MSD) corresponding to three representative order distributions: two-point, uniform, and beta. Notably, the uniform and beta distributions give rise to ultraslow diffusion. This study offers practical guidance for experimentalists seeking to apply FDEs with HP derivatives, emphasizing the importance of working within a physically consistent parameter regime. Our findings highlight the HP fractional framework as a powerful extension of classical FDEs, enabling a more versatile description of anomalous diffusion in systems with complex memory effects or trapping dynamics.