Special Issue "Fractional Differential Equations"

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: 20 December 2018

Special Issue Editor

Guest Editor
Prof. Dr. Gaston N'Guerekata

School of Computer, Mathematical and Natural Sciences, Morgan State University Baltimore, MD 21251, USA
Website | E-Mail
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

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Keywords

  • fractional derivative
  • fractional differential equations
  • fractional differential inclusions
  • existence of solutions
  • stability of solutions

Published Papers (4 papers)

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Research

Open AccessArticle Conformable Laplace Transform of Fractional Differential Equations
Received: 30 June 2018 / Revised: 31 July 2018 / Accepted: 4 August 2018 / Published: 7 August 2018
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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
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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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Open AccessArticle Lyapunov Functions to Caputo Fractional Neural Networks with Time-Varying Delays
Received: 29 March 2018 / Revised: 3 May 2018 / Accepted: 5 May 2018 / Published: 9 May 2018
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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
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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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Open AccessArticle Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives
Received: 19 December 2017 / Revised: 4 February 2018 / Accepted: 8 February 2018 / Published: 11 February 2018
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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
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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)
Open AccessArticle Mild Solutions to the Cauchy Problem for Some Fractional Differential Equations with Delay
Received: 5 October 2017 / Revised: 9 November 2017 / Accepted: 14 November 2017 / Published: 20 November 2017
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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
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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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