Distributed Order Calculus and Equations of Ultraslow Diffusion

TL;DR

This paper develops a mathematical framework for diffusion equations involving distributed order derivatives, which model ultraslow diffusion phenomena with logarithmic mean square displacement growth.

Contribution

It introduces a rigorous theory for distributed order derivatives and integrals, advancing the understanding of equations used in modeling ultraslow diffusion.

Findings

Established properties of distributed order derivatives and integrals.

Analyzed the mathematical structure of distributed order diffusion equations.

Provided tools for further analysis of ultraslow diffusion models.

Abstract

We consider diffusion type equations with a distributed order derivative in the time variable. This derivative is defined as the integral in $\alpha$ of the Caputo-Dzhrbashian fractional derivative of order $\alpha \in (0,1)$ with a certain positive weight function. Such equations are used in physical literature for modeling diffusion with a logarithmic growth of the mean square displacement. In this work we develop a mathematical theory of such equations, study the derivatives and integrals of distributed order.