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Fractal Fract
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Recent Advances in Fractional Evolution Equations and Related Topics
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Recent Advances in Fractional Evolution Equations and Related Topics

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 8012

Special Issue Editors

Prof. Dr. Gaston M. N'Guérékata

E-Mail Website
Guest Editor
School of Computer, Mathematical and Natural Sciences, Morgan State University, Baltimore, MD 21251, USA
Interests: functional analysis and abstract differential equations in Banach and locally convex spaces, with applications to partial differential equations and functional differential equations
Prof. Dr. Mouffak Benchohra

E-Mail Website
Guest Editor
Laboratory of Mathematics, Djillali Liabes University of Sidi Bel Abbes, P.O. Box 89, Sidi Bel Abbes 22000, Algeria
Interests: fractional differential equations; abstract differential equations; functional differential equations with delay
Special Issues, Collections and Topics in MDPI journals
Dr. Abdelkrim Salim

E-Mail Website
Guest Editor
Faculty of Technology, Hassiba Benbouali University, P.O. Box 151, Chlef 02000, Algeria
Interests: mathematical analysis; ordinary differential equations; functional analysis; differential equations; fractional differential equations; integral equations

Special Issue Information

Dear Colleagues,

The fractional order models of real systems are always more adequate than the classical integer order models, since the description of some systems is more accurate when the fractional derivative is used. The advantages of fractional differentiation have become evident in modeling the mechanical and electrical properties of real materials, description of rheological properties of rocks, and in various other fields. Such models are interesting for engineers and physicists as well as so-called pure mathematicians.

Fractional evolution equations provide a unifying framework to investigate the well-posedness of complex systems with fractional order derivatives. This Special Issue will present recent advances related to the existence, attractivity, stability, periodic solutions, and control theory for such classes of fractional evolution equations.

Prof. Dr. Gaston M. N'Guérékata
Prof. Dr. Mouffak Benchohra
Dr. Abdelkrim Salim
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

  • fractional evolution equation
  • mild solution
  • fixed point
  • Banach space
  • measure of noncompactness
  • semi-group of linear operators
  • controllability
  • stability

Published Papers (7 papers)

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Research

15 pages, 332 KiB  
Article
The Averaging Principle for Hilfer Fractional Stochastic Evolution Equations with Lévy Noise
by Min Yang, Ting Lv and Qiru Wang
Fractal Fract. 2023, 7(10), 701; https://doi.org/10.3390/fractalfract7100701 - 24 Sep 2023
Cited by 3 | Viewed by 760
Abstract
This article focuses on deriving the averaging principle for Hilfer fractional stochastic evolution equations (HFSEEs) driven by Lévy noise. We show that the solutions of the averaged equations converge to the corresponding solutions of the original equations, both in the sense of mean [...] Read more.
This article focuses on deriving the averaging principle for Hilfer fractional stochastic evolution equations (HFSEEs) driven by Lévy noise. We show that the solutions of the averaged equations converge to the corresponding solutions of the original equations, both in the sense of mean square and of probability. Our results enable us to focus on the averaged system rather than the original, more complex one. Given that the existing literature on the averaging principle for Hilfer fractional stochastic differential equations has been established in finite-dimensional spaces, the novelty here is the derivation of the averaging principle for a class of HFSEEs in Hilbert space. Furthermore, an example is allotted to illustrate the feasibility and utility of our results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Evolution Equations and Related Topics)
14 pages, 313 KiB  
Article
Approximate Controllability of Fractional Stochastic Evolution Inclusions with Non-Local Conditions
by Kinda Abuasbeh, Azmat Ullah Khan Niazi, Hafiza Maria Arshad, Muath Awadalla and Salma Trabelsi
Fractal Fract. 2023, 7(6), 462; https://doi.org/10.3390/fractalfract7060462 - 07 Jun 2023
Cited by 2 | Viewed by 829
Abstract
This article investigates the approximate controllability of non-linear fractional stochastic differential inclusions with non-local conditions. We establish a set of sufficient conditions for their approximate controllability and provide results in terms of controllability for the fractional stochastic control system. Our approach relies on [...] Read more.
This article investigates the approximate controllability of non-linear fractional stochastic differential inclusions with non-local conditions. We establish a set of sufficient conditions for their approximate controllability and provide results in terms of controllability for the fractional stochastic control system. Our approach relies on using fractional calculus and the fixed-point theorem for multiple-valued operators. Finally, we present an illustrative example to support our findings. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Evolution Equations and Related Topics)
16 pages, 345 KiB  
Article
On the Existence Results for a Mixed Hybrid Fractional Differential Equations of Sequential Type
by Meraa Arab, Muath Awadalla, Murugesan Manigandan, Kinda Abuasbeh, Nazim I. Mahmudov and Thangaraj Nandha Gopal
Fractal Fract. 2023, 7(3), 229; https://doi.org/10.3390/fractalfract7030229 - 04 Mar 2023
Cited by 3 | Viewed by 878
Abstract
In this article, we study the existence of a solution to the mixed hybrid fractional differential equations of sequential type with nonlocal integral hybrid boundary conditions. The main results are established with the aid of Darbo’s fixed point theorem and Hausdorff’s measure of [...] Read more.
In this article, we study the existence of a solution to the mixed hybrid fractional differential equations of sequential type with nonlocal integral hybrid boundary conditions. The main results are established with the aid of Darbo’s fixed point theorem and Hausdorff’s measure of noncompactness method. The stability of the proposed fractional differential equation is also investigated using the Ulam–Hyer technique. In addition, an applied example that supports the theoretical results reached through this study is included. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Evolution Equations and Related Topics)
15 pages, 308 KiB  
Article
A Study on the Approximate Controllability of Hilfer Fractional Evolution Systems
by Yue Liang
Fractal Fract. 2022, 6(12), 695; https://doi.org/10.3390/fractalfract6120695 - 24 Nov 2022
Viewed by 888
Abstract
In this article, the existence and uniqueness of mild solutions are investigated for Hilfer fractional evolution systems. Particularly, the approximate controllability is also investigated under some essential conditions by applying the sequence method. An example, as an application, is provided to demonstrate the [...] Read more.
In this article, the existence and uniqueness of mild solutions are investigated for Hilfer fractional evolution systems. Particularly, the approximate controllability is also investigated under some essential conditions by applying the sequence method. An example, as an application, is provided to demonstrate the obtained results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Evolution Equations and Related Topics)
27 pages, 378 KiB  
Article
Functional Integro-Differential Equations with State-Dependent Delay and Non-Instantaneous Impulsions: Existence and Qualitative Results
by Abdelhamid Bensalem, Abdelkrim Salim, Mouffak Benchohra and Gaston M. N’Guérékata
Fractal Fract. 2022, 6(10), 615; https://doi.org/10.3390/fractalfract6100615 - 21 Oct 2022
Cited by 10 | Viewed by 1206
Abstract
This paper addresses some existence, attractivity and controllability results for semilinear integrodifferential equations having non-instantaneous impulsions on an infinite interval via resolvent operators in case of neutral and state-dependent delay problems. Our criteria were obtained by applying a Darbo’s fixed-point theorem combined with [...] Read more.
This paper addresses some existence, attractivity and controllability results for semilinear integrodifferential equations having non-instantaneous impulsions on an infinite interval via resolvent operators in case of neutral and state-dependent delay problems. Our criteria were obtained by applying a Darbo’s fixed-point theorem combined with measures of noncompactness. The obtained result is illustrated by an example at the end. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Evolution Equations and Related Topics)
17 pages, 3964 KiB  
Article
A Variable-Order Fractional Constitutive Model to Characterize the Rate-Dependent Mechanical Behavior of Soft Materials
by Yunfei Gao, Deshun Yin and Bin Zhao
Fractal Fract. 2022, 6(10), 590; https://doi.org/10.3390/fractalfract6100590 - 13 Oct 2022
Cited by 5 | Viewed by 1187
Abstract
Building an accurate constitutive model for soft materials is essential for better understanding its rate-dependent deformation characteristics and improving the design of soft material devices. To establish a concise constitutive model with few parameters and clear physical meaning, a variable-order fractional model is [...] Read more.
Building an accurate constitutive model for soft materials is essential for better understanding its rate-dependent deformation characteristics and improving the design of soft material devices. To establish a concise constitutive model with few parameters and clear physical meaning, a variable-order fractional model is proposed to accurately describe and predict the rate-dependent mechanical behavior of soft materials. In this work, the discrete variable-order fractional operator enables the predicted stress response to be entirely consistent with the whole stress history and the fractional order’s path-dependent values. The proposed model is further implemented in a numerical form and applied to predict several typical soft materials’ tensile and compressive deformation behavior. Our research indicates that the proposed variable-order fractional constitutive model is capable of predicting the nonlinear rate-dependent mechanical behavior of soft materials with high accuracy and more convinced reliability in comparison with the existing fractional models, where the fractional order contains a constant initial order to depict the initial elastic response and a linear variable-order function to account for the strain hardening behavior after acrossing the yield point. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Evolution Equations and Related Topics)
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22 pages, 389 KiB  
Article
An Investigation on the Optimal Control for Hilfer Fractional Neutral Stochastic Integrodifferential Systems with Infinite Delay
by Murugesan Johnson and Velusamy Vijayakumar
Fractal Fract. 2022, 6(10), 583; https://doi.org/10.3390/fractalfract6100583 - 11 Oct 2022
Cited by 10 | Viewed by 1101
Abstract
The main concern of this manuscript is to study the optimal control problem for Hilfer fractional neutral stochastic integrodifferential systems with infinite delay. Initially, we establish the existence of mild solutions for the Hilfer fractional stochastic integrodifferential system with infinite delay via applying [...] Read more.
The main concern of this manuscript is to study the optimal control problem for Hilfer fractional neutral stochastic integrodifferential systems with infinite delay. Initially, we establish the existence of mild solutions for the Hilfer fractional stochastic integrodifferential system with infinite delay via applying fractional calculus, semigroups, stochastic analysis techniques, and the Banach fixed point theorem. In addition, we establish the existence of mild solutions of the Hilfer fractional neutral stochastic delay integrodifferential system. Further, we investigate the existence of optimal pairs for the Hilfer fractional neutral stochastic delay integrodifferential systems. We provide an illustration to clarify our results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Evolution Equations and Related Topics)
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