Fractional diffusion equation with distributed-order material derivative. Stochastic foundations

TL;DR

This paper develops the stochastic basis for fractional dynamics governed by distributed-order fractional material derivatives, introducing a Levy walk process whose scaling limit satisfies a corresponding fractional diffusion equation.

Contribution

It establishes the stochastic foundation for distributed-order fractional diffusion and derives the scaling limit process as a solution to the fractional diffusion equation.

Findings

The Levy walk process of distributed order is constructed.

The scaling limit process solves the fractional diffusion equation.

The probability density function of the limit process satisfies the equation in a weak sense.

Abstract

In this paper we present stochastic foundations of fractional dynamics driven by fractional material derivative of distributed order-type. Before stating our main result we present the stochastic scenario which underlies the dynamics given by fractional material derivative. Then we introduce a Levy walk process of distributed-order type to establish our main result, which is the scaling limit of the considered process. It appears that the probability density function of the scaling limit process fulfills, in a weak sense, the fractional diffusion equation with material derivative of distributed-order type.