Natural and modified forms of distributed-order fractional diffusion equations

We consider diffusion-like equations with time and space fractional derivatives of distributed-order for the kinetic description of anomalous diffusion and relaxation phenomena, whose mean squared displacement does not change as a power law in time. Correspondingly, the underlying processes cannot be viewed as self-affine random processes processing a unique Hurst exponent. We show that different forms of distributed-order equations, which we call “natural” and “modified” ones, serve as a useful tool to describe the processes which become more anomalous with time (retarding subdiffusion and accelerated superdiffusion) or less anomalous demonstrating the transition from anomalous to normal diffusion (accelerated subdiffusion and truncated Lévy flights). Fractional diffusion equation with the distributed-order time derivative also accounts for the logarithmic diffusion (strong anomaly).