Distributed order fractional sub-diffusion

TL;DR

This paper investigates distributed order fractional sub-diffusion equations, analyzing their solutions under various boundary conditions and revealing decay properties with bounds and examples for different diffusion parameters.

Contribution

It introduces solutions for distributed order fractional sub-diffusion equations with boundary conditions, highlighting decay behaviors and bounds for the functional dependence on diffusion parameters.

Findings

Functional decay properties are established.

Upper and lower bounds for the decay functional are computed.

Examples illustrate different decay rates.

Abstract

A distributed order fractional diffusion equation is considered. Distributed order derivatives are fractional derivatives that have been integrated over the order of the derivative within a given range. In this paper sub-diffusive cases are considered. That is, the order of the time derivative ranges from zero to one. The equation is solved for Dirichlet, Neumann, and Cauchy boundary conditions. The time dependence for each of the three cases is found to be a functional of the diffusion parameter. This functional is shown to have decay properties. Upper and lower bounds are computed for the functional. Examples are also worked out for comparative decay rates.