Distributed order time fractional diffusion equation. (English) Zbl 1089.60046

The authors present the basic theory of time fractional diffusion with distributed orders (between 0 and 1) of the temporal derivative. If there is more than one order present, the density corresponding to the fundamental solution (with the delta function as initial condition) evolves as a kind of retarded subdiffusion (its evolving second moment evolving like time raised to a decreasing power). The authors illustrate such behaviour by plots of results of simulations by a backward oriented approximating random walk based on the Grünwald-Letnikov approximation. Furthermore they show that by special choice of the order density ultra-slow diffusion is otained (in which the second moment grows like a power of the logarithm of time). They outline the subordination to a Wiener process and hint at the relation to the theory of continuous time random walks.


60J60 Diffusion processes
26A33 Fractional derivatives and integrals
45K05 Integro-partial differential equations
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics