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Fractal Fract
Special Issues
Probabilistic Solutions and Stochastic Representation of Fractional Differential Equations
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Probabilistic Solutions and Stochastic Representation of Fractional Differential Equations

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

Deadline for manuscript submissions: closed (17 August 2022) | Viewed by 8598

Special Issue Editors

Dr. Giacomo Ascione

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Guest Editor
Department of Mathematics and Applications "R.Caccioppoli", Università degli Studi di Napoli Federico II, 80126 Naples, Italy
Interests: generalized fractional calculus; lévy process; bernstein function; subordinator; nonlocal abstract Cauchy problems; subordinated Brownian motion
Dr. Alessandra Meoli

E-Mail Website
Guest Editor
Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy
Interests: probability and mathematical statistics
Prof. Dr. Enrica Pirozzi

E-Mail Website
Guest Editor
Dipartimento di Matematica e Applicazioni, University of Naples FEDERICO II, Via Cinthia, Monte S. Angelo, 80126 Napoli, Italy
Interests: stochastic processes; gaussian processes; first passage problem; fractional stochastic processes; diffusion processes; fractional brownian motion; queueing theory; stochastic time changes; neuronal models

Special Issue Information

Dear Colleagues,

Fractional differential equations have been shown to be an important tool in probability due to their strict connection with some classes of processes that are widely used for modeling purposes. With this connection, different stochastic representation results have arisen, which are helpful in a twofold way: on one hand, one can use fractional equations (and both analytical and numerical methods for them) to determine some characteristics of the considered process; on the other hand, fractional equations gain a more visualizable interpretation (that was already useful in the classical case) that can help to determine some properties of their solutions. This interplay is mirrored in the modeling context as fractional differential equations are used to represent macroscopic behavior whose microscopic explanation is given by the involved stochastic processes. For these reasons, finding probabilistic solutions to (possibly generalized) fractional differential equations improves both our knowledge on the equation and the involved process, thus leading to an improvement of the modeling techniques involving fractional operators. Potential topics include but are not limited to:

  • Stochastic representation results for solutions of time/space-fractional ordinary and partial differential equations;
  • Probabilistic solutions to distributed-order time/space-fractional ordinary and partial differential equations;
  • Generalized fractional calculus, time-nonlocal equations, and stochastic processes;
  • Bernstein functions of elliptic operators and generators of jump processes;
  • Fractional order stochastic (partial) differential equations;
  • Deterministic and stochastic modeling involving fractional order operators;
  • Numerical methods for fractional (partial) differential equations.

Dr. Giacomo Ascione
Dr. Alessandra Meoli
Prof. Dr. Enrica Pirozzi
Guest Editors

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

  • stochastic processes
  • fractional order differential operators
  • distributed order differential operators
  • fractional calculus
  • generalized fractional calculus
  • time/space-nonlocal equations
  • bernstein functions
  • jump processes
  • semi-markov processes
  • stochastic modeling
  • numerical methods for fractional differential equations
  • stochastic simulation

Published Papers (5 papers)

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Research

15 pages, 331 KiB  
Article
Hermite–Hadamard Type Inequalities Involving (k-p) Fractional Operator for Various Types of Convex Functions
by Vuk Stojiljković, Rajagopalan Ramaswamy, Fahad Alshammari, Ola A. Ashour, Mohammed Lahy Hassan Alghazwani and Stojan Radenović
Fractal Fract. 2022, 6(7), 376; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6070376 - 02 Jul 2022
Cited by 17 | Viewed by 1431
Abstract
We establish various fractional convex inequalities of the Hermite–Hadamard type with addition to many other inequalities. Various types of such inequalities are obtained, such as (p,h) fractional type inequality and many others, as the (p,h) [...] Read more.
We establish various fractional convex inequalities of the Hermite–Hadamard type with addition to many other inequalities. Various types of such inequalities are obtained, such as (p,h) fractional type inequality and many others, as the (p,h)-convexity is the generalization of the other convex inequalities. As a consequence of the (h,m)-convexity, the fractional inequality of the (s,m)-type is obtained. Many consequences of such fractional inequalities and generalizations are obtained. Full article
(This article belongs to the Special Issue Probabilistic Solutions and Stochastic Representation of Fractional Differential Equations)
19 pages, 6497 KiB  
Article
Consensus of Fractional-Order Double-Integral Multi-Agent System in a Bounded Fluctuating Potential
by Xi Chen, Maokang Luo and Lu Zhang
Fractal Fract. 2022, 6(3), 147; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6030147 - 07 Mar 2022
Cited by 2 | Viewed by 1621
Abstract
At present, the consensus problem of fractional complex systems has received more attention. However, there is little literature on the consensus problem of fractional-order complex systems under noise disturbance. In this paper, we present a fractional-order double-integral multi-agent system affected by a common [...] Read more.
At present, the consensus problem of fractional complex systems has received more attention. However, there is little literature on the consensus problem of fractional-order complex systems under noise disturbance. In this paper, we present a fractional-order double-integral multi-agent system affected by a common bounded fluctuating potential, where the protocol term consists of both the relative position and velocity information of neighboring agents. The consensus conditions of the presented system in the absence of noise are analytically given and verified by a numerical simulation algorithm. Then, the influences of the system order and other system parameters on the consensus of the presented system in the presence of bounded noise are also analyzed. It is found that when compared with the classical integer-order system, the presented fractional-order system has a larger range of consensus parameters and has more rich dynamic characteristics under the action of random noise. Especially, the bounded noise has a promoting effect on the consensus of the presented fractional-order system, while there is no similar phenomenon in the corresponding integer-order system. Full article
(This article belongs to the Special Issue Probabilistic Solutions and Stochastic Representation of Fractional Differential Equations)
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12 pages, 291 KiB  
Article
Blow-Up of Solutions to Fractional-in-Space Burgers-Type Equations
by Munirah Alotaibi, Mohamed Jleli and Bessem Samet
Fractal Fract. 2021, 5(4), 249; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040249 - 01 Dec 2021
Cited by 1 | Viewed by 1574
Abstract
We consider fractional-in-space analogues of Burgers equation and Korteweg-de Vries-Burgers equation on bounded domains. Namely, we establish sufficient conditions for finite-time blow-up of solutions to the mentioned equations. The obtained conditions depend on the initial value and the boundary conditions. Some examples are [...] Read more.
We consider fractional-in-space analogues of Burgers equation and Korteweg-de Vries-Burgers equation on bounded domains. Namely, we establish sufficient conditions for finite-time blow-up of solutions to the mentioned equations. The obtained conditions depend on the initial value and the boundary conditions. Some examples are provided to illustrate our obtained results. In the proofs of our main results, we make use of the test function method and some integral inequalities. Full article
(This article belongs to the Special Issue Probabilistic Solutions and Stochastic Representation of Fractional Differential Equations)
19 pages, 351 KiB  
Article
Mixed Caputo Fractional Neutral Stochastic Differential Equations with Impulses and Variable Delay
by Mahmoud Abouagwa, Rashad A. R. Bantan, Waleed Almutiry, Anas D. Khalaf and Mohammed Elgarhy
Fractal Fract. 2021, 5(4), 239; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040239 - 23 Nov 2021
Cited by 8 | Viewed by 1528
Abstract
In this manuscript, a new class of impulsive fractional Caputo neutral stochastic differential equations with variable delay (IFNSDEs, in short) perturbed by fractional Brownain motion (fBm) and Poisson jumps was studied. We utilized the Carathéodory approximation approach and stochastic calculus to present the [...] Read more.
In this manuscript, a new class of impulsive fractional Caputo neutral stochastic differential equations with variable delay (IFNSDEs, in short) perturbed by fractional Brownain motion (fBm) and Poisson jumps was studied. We utilized the Carathéodory approximation approach and stochastic calculus to present the existence and uniqueness theorem of the stochastic system under Carathéodory-type conditions with Lipschitz and non-Lipschitz conditions as special cases. Some existing results are generalized and enhanced. Finally, an application is offered to illustrate the obtained theoretical results. Full article
(This article belongs to the Special Issue Probabilistic Solutions and Stochastic Representation of Fractional Differential Equations)
11 pages, 308 KiB  
Article
An Infinite System of Fractional Order with p-Laplacian Operator in a Tempered Sequence Space via Measure of Noncompactness Technique
by Ahmed Salem, Lamya Almaghamsi and Faris Alzahrani
Fractal Fract. 2021, 5(4), 182; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040182 - 25 Oct 2021
Cited by 12 | Viewed by 1406
Abstract
In the current study, a new class of an infinite system of two distinct fractional orders with p-Laplacian operator is presented. Our mathematical model is introduced with the Caputo–Katugampola fractional derivative which is considered a generalization to the Caputo and Hadamard fractional [...] Read more.
In the current study, a new class of an infinite system of two distinct fractional orders with p-Laplacian operator is presented. Our mathematical model is introduced with the Caputo–Katugampola fractional derivative which is considered a generalization to the Caputo and Hadamard fractional derivatives. In a new sequence space associated with a tempered sequence and the sequence space c0 (the space of convergent sequences to zero), a suitable new Hausdorff measure of noncompactness form is provided. This formula is applied to discuss the existence of a solution to our infinite system through applying Darbo’s theorem which extends both the classical Banach contraction principle and the Schauder fixed point theorem. Full article
(This article belongs to the Special Issue Probabilistic Solutions and Stochastic Representation of Fractional Differential Equations)
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