Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions

Abstract

In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is then discussed in detail. The ultra-slow behaviour emerges from the explicit form of the variance of the random process arising from our analysis. Finally, we obtain a particular solution for the nonlinear Hadamard-diffusive equation.

A numerical approximation for the Caputo-Hadamard derivative and its application in time-fractional variable-coefficient diffusion equation

“…where x + i−1/2 = lim →0 + (x i−1/2 + ), y + j−1/2 = lim →0 + (y j−1/2 + ), and n is the outward unit normal vector of the integration region. Furthermore, the projections have the following approximation properties [6].…”

Blowup for semilinear fractional diffusion system with Caputo–Hadamard derivative

“…The Hadamard-type fractional calculus operators have also been used to describe the ultraslowly diffusive process, e.g., Sinai diffusion, 5 or the Lomnitz logarithmic creep law such as igneous rock. 6 More information on the topic can be referred to Denisov and Kantz, de Gregorio and Garra, and Schiessel et al [7][8][9] or recent review. 10 Some studies related to Hadamard-type fractional differential equations can be found in Kilbas and Li and Li [11][12][13][14][15] and the references cited therein.…”