Fractional Cauchy Problems on Bounded Domains: Survey of Recent Results

Abstract

In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. This problem was first considered by Nigmatullin [25] (Nigmatullin RR (1986) The realization of the generalized transfer in a medium with fractal geometry. Phys Status Solidi B 133:425–430), and Zasalavsky [32] (Zaslavsky G (1994) Fractional kinetic equation for Hamiltonian chaos. Phys D 76:110–122) in ${\mathbb{R}}^{d}$ for modeling some physical phenomena. The fractional derivative models time delays in a diffusion process. We will give a survey of the recent results on the fractional Cauchy problem and its generalizations on bounded domains D⊂ℝ ^d obtained in Meerschaert et al.[20, 21](Meerschaert MM, Nane E, Vellaisamy P (2009) Fractional Cauchy problems on bounded domains. Ann Probab 37:979–1007; Meerschaert MM, Nane E, Vellaisamy P (to appear) Distributed-order fractional diffusions on bounded domains. J Math Anal Appl). We also study the solutions of fractional Cauchy problem where the first time derivative is replaced with an infinite sum of fractional derivatives. We point out connections to eigenvalue problems for the fractional time operators considered. The solutions to the eigenvalue problems are expressed by Mittag-Leffler functions and its generalized versions. The stochastic solution of the eigenvalue problems for the fractional derivatives are given by inverse subordinators.