Strong analytic solutions of fractional Cauchy problems

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.