Distributed Order Calculus and Equations of Ultraslow Diffusion

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.

39 pages. To appear in J. Math. Anal. Appl

References in corpus (3)

• Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations
• Distributed-Order Fractional Kinetics
• Fractional Fokker-Planck Equation for Ultraslow Kinetics

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