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Fractal Fract
Special Issues
Fractional and Anomalous Diffusions on Regular and Irregular Domains
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Fractional and Anomalous Diffusions on Regular and Irregular Domains

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (30 June 2021) | Viewed by 7333

Special Issue Editor

Prof. Dr. Mirko D'Ovidio

E-Mail Website
Guest Editor
Sapienza University of Rome, Rome, Italy
Interests: PDEs; fractional calculus; boundary value problems; trace processes; multiplicative functionals; time changes

Special Issue Information

Dear Colleagues,

Anomalous behavior can be regarded as the common property of a wide class of phenomena. Such a class includes motions driven by fractional equations and motions on irregular (or non-homogeneous) domains. Sometimes, this respectively agrees with the macroscopic or microscopic analysis of real phenomena. Indeed, in macroscopic analysis, we bear the fact that a motion exhibits its own anomalous dynamic; that is, the motion can be written as a time change of a base process where the governing equation is a fractional partial differential equation. The microscopic analysis aims to relate the anomalous behavior of the motion with the geometry of the medium (boundaries, interfaces, layers, etc.). The applications of fractional calculus in many fields of applied sciences attracted the increasing interest of many researchers in recent years. This Special Issue aims to collect recent perspectives in fractional calculus, applied to all problems arising in all fields of science, engineering applications, and other applied fields.

Prof. Dr. Mirko D'Ovidio
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematics
  • physics
  • mathematical physics
  • probability
  • fractals
  • fractional calculus
  • fractional powers of operators
  • nonlocal operators

Published Papers (4 papers)

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Research

17 pages, 340 KiB  
Article
Lévy Processes Linked to the Lower-Incomplete Gamma Function
by Luisa Beghin and Costantino Ricciuti
Fractal Fract. 2021, 5(3), 72; https://doi.org/10.3390/fractalfract5030072 - 17 Jul 2021
Cited by 2 | Viewed by 1582
Abstract
We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to [...] Read more.
We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior. Full article
(This article belongs to the Special Issue Fractional and Anomalous Diffusions on Regular and Irregular Domains)
9 pages, 279 KiB  
Article
Approximation of Space-Time Fractional Equations
by Raffaela Capitanelli and Mirko D’Ovidio
Fractal Fract. 2021, 5(3), 71; https://doi.org/10.3390/fractalfract5030071 - 17 Jul 2021
Viewed by 1284
Abstract
The aim of this paper is to provide approximation results for space-time non-local equations with general non-local (and fractional) operators in space and time. We consider a general Markov process time changed with general subordinators or inverses to general subordinators. Our analysis is [...] Read more.
The aim of this paper is to provide approximation results for space-time non-local equations with general non-local (and fractional) operators in space and time. We consider a general Markov process time changed with general subordinators or inverses to general subordinators. Our analysis is based on Bernstein symbols and Dirichlet forms, where the symbols characterize the time changes, and the Dirichlet forms characterize the Markov processes. Full article
(This article belongs to the Special Issue Fractional and Anomalous Diffusions on Regular and Irregular Domains)
12 pages, 273 KiB  
Article
Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions
by Alessandro De Gregorio and Roberto Garra
Fractal Fract. 2021, 5(2), 48; https://doi.org/10.3390/fractalfract5020048 - 23 May 2021
Cited by 7 | Viewed by 2011
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional and Anomalous Diffusions on Regular and Irregular Domains)
21 pages, 358 KiB  
Article
Approximate Controllability of Fully Nonlocal Stochastic Delay Control Problems Driven by Hybrid Noises
by Lixu Yan and Yongqiang Fu
Fractal Fract. 2021, 5(2), 30; https://doi.org/10.3390/fractalfract5020030 - 12 Apr 2021
Cited by 3 | Viewed by 1599
Abstract
In this paper, a class of time-space fractional stochastic delay control problems with fractional noises and Poisson jumps in a bounded domain is considered. The proper function spaces and assumptions are proposed to discuss the existence of mild solutions. In particular, approximate strategy [...] Read more.
In this paper, a class of time-space fractional stochastic delay control problems with fractional noises and Poisson jumps in a bounded domain is considered. The proper function spaces and assumptions are proposed to discuss the existence of mild solutions. In particular, approximate strategy is used to obtain the existence of mild solutions for the problem with linear fractional noises; fixed point theorem is used to achieve the existence of mild solutions for the problem with nonlinear fractional noises. Finally, the approximate controllability of the problems with linear and nonlinear fractional noises is proved by the property of mild solutions. Full article
(This article belongs to the Special Issue Fractional and Anomalous Diffusions on Regular and Irregular Domains)
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