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Axioms
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Fractional Differential Equations
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Fractional Differential Equations

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (20 December 2018) | Viewed by 34919

Special Issue Editor

Prof. Dr. Gaston M. N'Guerekata

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

Special Issue Information

Dear Colleagues,

During the last decade, there has been an increased interest in fractional dynamics, as it was found that to play a fundamental role in the modeling of numerous phenomena, in particular, complex media, and long-memory media, or porous media. The aim of this Special Issue is to present recent developments in the theory, analytic as well as numerical. Potential topics include but are not limited to existence, stability, oscillatory and asymptotic behavior of solutions, numerical results, and applications in sciences and engineering.

Prof. Dr. Gaston N'Guerekata
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. Axioms 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 2400 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 derivative
  • fractional differential equations
  • fractional differential inclusions
  • existence of solutions
  • stability of solutions

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Published Papers (7 papers)

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Research

16 pages, 346 KiB  
Article
Operator Ordering and Solution of Pseudo-Evolutionary Equations
by Nicolas Behr, Giuseppe Dattoli and Ambra Lattanzi
Axioms 2019, 8(1), 35; https://doi.org/10.3390/axioms8010035 - 16 Mar 2019
Cited by 1 | Viewed by 3188
Abstract
The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and [...] Read more.
The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain time- and operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions for fractional differential equations. We apply the method to several examples, in which fractional calculus and a certain umbral image calculus play a role of central importance. Full article
(This article belongs to the Special Issue Fractional Differential Equations)
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15 pages, 336 KiB  
Article
Harrod–Domar Growth Model with Memory and Distributed Lag
by Vasily E. Tarasov and Valentina V. Tarasova
Axioms 2019, 8(1), 9; https://doi.org/10.3390/axioms8010009 - 15 Jan 2019
Cited by 12 | Viewed by 8438
Abstract
In this paper, we propose a macroeconomic growth model, in which we take into account memory with power-law fading and gamma distributed lag. This model is a generalization of the standard Harrod–Domar growth model. Fractional differential equations of this generalized model with memory [...] Read more.
In this paper, we propose a macroeconomic growth model, in which we take into account memory with power-law fading and gamma distributed lag. This model is a generalization of the standard Harrod–Domar growth model. Fractional differential equations of this generalized model with memory and lag are suggested. For these equations, we obtain solutions, which describe the macroeconomic growth of national income with fading memory and distributed time-delay. The asymptotic behavior of these solutions is described. Full article
(This article belongs to the Special Issue Fractional Differential Equations)
13 pages, 307 KiB  
Article
Regional Enlarged Observability of Fractional Differential Equations with Riemann—Liouville Time Derivatives
by Hayat Zouiten, Ali Boutoulout and Delfim F. M. Torres
Axioms 2018, 7(4), 92; https://doi.org/10.3390/axioms7040092 - 1 Dec 2018
Cited by 2 | Viewed by 2919
Abstract
We introduce the concept of regional enlarged observability for fractional evolution differential equations involving Riemann–Liouville derivatives. The Hilbert Uniqueness Method (HUM) is used to reconstruct the initial state between two prescribed functions, in an interested subregion of the whole domain, without the knowledge [...] Read more.
We introduce the concept of regional enlarged observability for fractional evolution differential equations involving Riemann–Liouville derivatives. The Hilbert Uniqueness Method (HUM) is used to reconstruct the initial state between two prescribed functions, in an interested subregion of the whole domain, without the knowledge of the state. Full article
(This article belongs to the Special Issue Fractional Differential Equations)
12 pages, 289 KiB  
Article
Conformable Laplace Transform of Fractional Differential Equations
by Fernando S. Silva, Davidson M. Moreira and Marcelo A. Moret
Axioms 2018, 7(3), 55; https://doi.org/10.3390/axioms7030055 - 7 Aug 2018
Cited by 72 | Viewed by 7211
Abstract
In this paper, we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant [...] Read more.
In this paper, we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, the analytical solution for a class of fractional models associated with the logistic model, the von Foerster model and the Bertalanffy model is presented graphically for various fractional orders. The solution of the corresponding classical model is recovered as a particular case. Full article
(This article belongs to the Special Issue Fractional Differential Equations)
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18 pages, 8405 KiB  
Article
Lyapunov Functions to Caputo Fractional Neural Networks with Time-Varying Delays
by Ravi Agarwal, Snezhana Hristova and Donal O’Regan
Axioms 2018, 7(2), 30; https://doi.org/10.3390/axioms7020030 - 9 May 2018
Cited by 3 | Viewed by 3696
Abstract
One of the main properties of solutions of nonlinear Caputo fractional neural networks is stability and often the direct Lyapunov method is used to study stability properties (usually these Lyapunov functions do not depend on the time variable). In connection with the Lyapunov [...] Read more.
One of the main properties of solutions of nonlinear Caputo fractional neural networks is stability and often the direct Lyapunov method is used to study stability properties (usually these Lyapunov functions do not depend on the time variable). In connection with the Lyapunov fractional method we present a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. These derivatives are applied to various types of neural networks with variable coefficients and time-varying delays. We show that quadratic Lyapunov functions and their Caputo fractional derivatives are not applicable in some cases when one studies stability properties. Some sufficient conditions for stability of equilibrium of nonlinear Caputo fractional neural networks with time dependent transmission delays, time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. The cases of time varying Lipschitz coefficients as well as nonLipschitz activation functions are studied. We illustrate our theory on particular nonlinear Caputo fractional neural networks. Full article
(This article belongs to the Special Issue Fractional Differential Equations)
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18 pages, 256 KiB  
Article
Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives
by Jun Jiang, Yuqiang Feng and Shougui Li
Axioms 2018, 7(1), 10; https://doi.org/10.3390/axioms7010010 - 11 Feb 2018
Cited by 19 | Viewed by 4890
Abstract
In this paper, the solvability of nonlinear fractional partial differential equations (FPDEs) with mixed partial derivatives is considered. The invariant subspace method is generalized and is then used to derive exact solutions to the nonlinear FPDEs. Some examples are solved to illustrate the [...] Read more.
In this paper, the solvability of nonlinear fractional partial differential equations (FPDEs) with mixed partial derivatives is considered. The invariant subspace method is generalized and is then used to derive exact solutions to the nonlinear FPDEs. Some examples are solved to illustrate the effectiveness and applicability of the method. Full article
(This article belongs to the Special Issue Fractional Differential Equations)
232 KiB  
Article
Mild Solutions to the Cauchy Problem for Some Fractional Differential Equations with Delay
by Jin Liang and Yunyi Mu
Axioms 2017, 6(4), 30; https://doi.org/10.3390/axioms6040030 - 20 Nov 2017
Cited by 4 | Viewed by 3678
Abstract
In this paper, we present new existence theorems of mild solutions to Cauchy problem for some fractional differential equations with delay. Our main tools to obtain our results are the theory of analytic semigroups and compact semigroups, the Kuratowski measure of non-compactness, and [...] Read more.
In this paper, we present new existence theorems of mild solutions to Cauchy problem for some fractional differential equations with delay. Our main tools to obtain our results are the theory of analytic semigroups and compact semigroups, the Kuratowski measure of non-compactness, and fixed point theorems, with the help of some estimations. Examples are also given to illustrate the applicability of our results. Full article
(This article belongs to the Special Issue Fractional Differential Equations)
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