Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions

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.

Fractional Cauchy problem on random snowflakes

We consider time-changed Brownian motions on random Koch (pre-fractal and fractal) domains where the time change is given by the inverse to a subordinator. In particular, we study the fractional Cauchy problem with Robin condition on the pre-fractal boundary obtaining asymptotic results for the corresponding fractional diffusions with Robin, Neumann and Dirichlet boundary conditions on the fractal domain.

Retraction Note: Existence of global solutions for the semilinear nonlocal fractional Cauchy problem of the Schrödinger equation

This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1186/s13661-021-01488-8

Fuzzy Conformable Fractional Semigroups of Operators

In this paper, we introduce a fuzzy fractional semigroup of operators whose generator will be the fuzzy fractional derivative of the fuzzy semigroup at t = 0 . We establish some of their proprieties and some results about the solution of fuzzy fractional Cauchy problem.