Strong analytic solutions of fractional Cauchy problems

TL;DR

This paper develops strong analytic solutions for distributed order fractional Cauchy problems, extending previous single-order results to model complex diffusion processes with multiple delay sources.

Contribution

It extends existing fractional Cauchy problem solutions from single-order to distributed order derivatives, enabling more accurate modeling of complex diffusion phenomena.

Findings

Derived explicit solutions for distributed order fractional Cauchy problems

Extended the theory of fractional derivatives to include distributed orders

Provided mathematical framework for modeling ultra-slow diffusion processes

Abstract

Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. In some special cases distributed order derivative can be used to model ultra-slow diffusion. We extend the results of (Baeumer, B. and Meerschaert, M. M. Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4 (2001), 481--500.) in the single order fractional derivative case to distributed order fractional derivative case. In particular, we develop the strong analytic solutions of distributed order fractional Cauchy problems.