Space-time duality and high-order fractional diffusion

Space-time duality for fractional diffusion

Zolotarev proved a duality result that relates stable densities with different indices. In this paper, we show how Zolotarev duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivatives in place of the usual integer order derivatives. They govern scaling limits of random walk models, with power law jumps leading to fractional de...

Solution to time fractional generalized KdV of order 2q+1 and system of space fractional PDEs

Abstract. In this work, it has been shown that the combined use of exponential operators and integral transforms provides a powerful tool to solve time fractional generalized KdV of order 2q+1 and certain fractional PDEs. It is shown that exponential operators are an effective method for solving certain fractional linear equations with non-constant coefficients. It may be concluded that the com...

Time-fractional Diffusion of Distributed Order

The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general distribution of time orders we provide the fundamental solution, that is still a probability density, in terms of an integral of Laplace type. The kernel depends on the typ...

Space–time fractional diffusion on bounded domains

Dependence of Fractional Order Diffusion Model Parameters on Diffusion Time

INTRODUCTION At high b-values (e.g., ≥1,500 s/mm), it is well known that diffusion-induced MR signal loss in the brain tissue exhibits anomalous behavior that cannot be described by a mono-exponential function. This phenomenon has been attributed to tissue heterogeneity manifested by cellular structures, cell membranes, and/or intraand extra-cellular spaces [1]. Over the past few years, a numbe...