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Fractal Fract
Volume 5
Issue 2
10.3390/fractalfract5020048
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Article

Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions

by
Alessandro De Gregorio
* and
Roberto Garra
*
Department of Statistical Sciences, Sapienza, University of Rome, P.le Aldo Moro, 5, 00185 Roma, Italy
*
Authors to whom correspondence should be addressed.
Fractal Fract. 2021, 5(2), 48; https://doi.org/10.3390/fractalfract5020048
Submission received: 19 April 2021 / Revised: 18 May 2021 / Accepted: 20 May 2021 / Published: 23 May 2021
(This article belongs to the Special Issue Fractional and Anomalous Diffusions on Regular and Irregular Domains)
Versions Notes

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.
Keywords: anomalous diffusions; Hadamard fractional derivatives; inverse stable subordinators; Lévy processes anomalous diffusions; Hadamard fractional derivatives; inverse stable subordinators; Lévy processes

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MDPI and ACS Style

De Gregorio, A.; Garra, R. Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions. Fractal Fract. 2021, 5, 48. https://doi.org/10.3390/fractalfract5020048

AMA Style

De Gregorio A, Garra R. Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions. Fractal and Fractional. 2021; 5(2):48. https://doi.org/10.3390/fractalfract5020048

Chicago/Turabian Style

De Gregorio, Alessandro, and Roberto Garra. 2021. "Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions" Fractal and Fractional 5, no. 2: 48. https://doi.org/10.3390/fractalfract5020048

APA Style

De Gregorio, A., & Garra, R. (2021). Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions. Fractal and Fractional, 5(2), 48. https://doi.org/10.3390/fractalfract5020048

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