Fractional Differential Equations: Theory, Methods and Applications

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (28 February 2019) | Viewed by 30654

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Department of Statistics, Mathematical Analysis and Optimization, CITMAga, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain
Interests: nonlinear differential equations; fractional models; biomedical applications; digital twins
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Interests: ordinary and functional differential equations; fractional differential equations; nonlinear analysis; fixed point theory; fuzzy mathematics

Special Issue Information

Dear Colleagues,

Fractional calculus provides the possibility of introducing integrals and derivatives of arbitrary order in the mathematical modelling of physical processes, and it has become a relevant subject with applications to various fields, such as anomalous diffusion, propagation in different media or in relation with materials with different properties. However, many aspects from the theoretical and practical point of view have still to be developed in relation with models based on fractional operators.

This Special Issue is related to the new developments on different aspects of fractional differential equations, both from the theoretical point of view and from the applications in different fields such as physics, chemistry, or control theory, for instance. The topics of the issue include: fractional calculus, the mathematical analysis of the properties of the solutions to fractional equations, the extension of classical approaches, or the applications of fractional equations to several fields, among others.

Prof. Dr. Juan J. Nieto
Prof. Rosana Rodríguez-López
Guest Editors

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Keywords

  • Fractional Calculus
  • Equations with Fractional Operators
  • Fractional Differential Equations
  • Applications

Published Papers (11 papers)

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Research

13 pages, 264 KiB  
Article
On Conformable Double Laplace Transform and One Dimensional Fractional Coupled Burgers’ Equation
by Hassan Eltayeb, Imed Bachar and Adem Kılıçman
Symmetry 2019, 11(3), 417; https://doi.org/10.3390/sym11030417 - 21 Mar 2019
Cited by 17 | Viewed by 2628
Abstract
In the present work we introduced a new method and name it the conformable double Laplace decomposition method to solve one dimensional regular and singular conformable functional Burger’s equation. We studied the existence condition for the conformable double Laplace transform. In order to [...] Read more.
In the present work we introduced a new method and name it the conformable double Laplace decomposition method to solve one dimensional regular and singular conformable functional Burger’s equation. We studied the existence condition for the conformable double Laplace transform. In order to obtain the exact solution for nonlinear fractional problems, then we modified the double Laplace transform and combined it with the Adomian decomposition method. Later, we applied the new method to solve regular and singular conformable fractional coupled Burgers’ equations. Further, in order to illustrate the effectiveness of present method, we provide some examples. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)
12 pages, 242 KiB  
Article
Even Higher Order Fractional Initial Boundary Value Problem with Nonlocal Constraints of Purely Integral Type
by Said Mesloub and Faten Aldosari
Symmetry 2019, 11(3), 305; https://doi.org/10.3390/sym11030305 - 01 Mar 2019
Cited by 8 | Viewed by 2071
Abstract
In this paper, the a priori estimate method, the so-called energy inequalities method based on some functional analysis tools is developed for a Caputo time fractional 2 m th order diffusion wave equation with purely nonlocal conditions of integral type. Existence and uniqueness [...] Read more.
In this paper, the a priori estimate method, the so-called energy inequalities method based on some functional analysis tools is developed for a Caputo time fractional 2 m th order diffusion wave equation with purely nonlocal conditions of integral type. Existence and uniqueness of the solution are proved. The proofs of the results are based on some a priori estimates and on some density arguments. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)
15 pages, 838 KiB  
Article
Balanced Truncation Model Order Reduction in Limited Frequency and Time Intervals for Discrete-Time Commensurate Fractional-Order Systems
by Marek Rydel, Rafał Stanisławski and Krzysztof J. Latawiec
Symmetry 2019, 11(2), 258; https://doi.org/10.3390/sym11020258 - 19 Feb 2019
Cited by 6 | Viewed by 2137
Abstract
In this paper we investigate an implementation of new model order reduction techniques to linear time-invariant discrete-time commensurate fractional-order state space systems to obtain lower dimensional fractional-order models. Since the models of physical systems correctly approximate the physical phenomena of the modeled systems [...] Read more.
In this paper we investigate an implementation of new model order reduction techniques to linear time-invariant discrete-time commensurate fractional-order state space systems to obtain lower dimensional fractional-order models. Since the models of physical systems correctly approximate the physical phenomena of the modeled systems for restricted time and frequency ranges only, a special attention is given to time- and frequency-limited balanced truncation and frequency-weighted methods. Mathematical formulas for calculation of the time- and frequency-limited, as well as frequency-weighted controllability and observability Gramians, are extended to fractional-order systems. An instructive simulation experiment corroborates the potential of the introduced methodology. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)
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15 pages, 1063 KiB  
Article
Laplace Adomian Decomposition Method for Multi Dimensional Time Fractional Model of Navier-Stokes Equation
by Shahid Mahmood, Rasool Shah, Hassan khan and Muhammad Arif
Symmetry 2019, 11(2), 149; https://doi.org/10.3390/sym11020149 - 29 Jan 2019
Cited by 47 | Viewed by 4235
Abstract
In this research paper, a hybrid method called Laplace Adomian Decomposition Method (LADM) is used for the analytical solution of the system of time fractional Navier-Stokes equation. The solution of this system can be obtained with the help of Maple software, which provide [...] Read more.
In this research paper, a hybrid method called Laplace Adomian Decomposition Method (LADM) is used for the analytical solution of the system of time fractional Navier-Stokes equation. The solution of this system can be obtained with the help of Maple software, which provide LADM algorithm for the given problem. Moreover, the results of the proposed method are compared with the exact solution of the problems, which has confirmed, that as the terms of the series increases the approximate solutions are convergent to the exact solution of each problem. The accuracy of the method is examined with help of some examples. The LADM, results have shown that, the proposed method has higher rate of convergence as compare to ADM and HPM. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)
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11 pages, 265 KiB  
Article
On Some Fractional Integral Inequalities of Hermite-Hadamard’s Type through Convexity
by Shahid Qaisar, Jamshed Nasir, Saad Ihsan Butt and Sabir Hussain
Symmetry 2019, 11(2), 137; https://doi.org/10.3390/sym11020137 - 26 Jan 2019
Cited by 7 | Viewed by 2641
Abstract
In this paper, we incorporate the notion of convex function and establish new integral inequalities of type Hermite–Hadamard via Riemann—Liouville fractional integrals. It is worth mentioning that the obtained inequalities generalize Hermite–Hadamard type inequalities presented by Özdemir, M.E. et. al. (2013) and Sarikaya, [...] Read more.
In this paper, we incorporate the notion of convex function and establish new integral inequalities of type Hermite–Hadamard via Riemann—Liouville fractional integrals. It is worth mentioning that the obtained inequalities generalize Hermite–Hadamard type inequalities presented by Özdemir, M.E. et. al. (2013) and Sarikaya, M.Z. et. al. (2011). Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)
9 pages, 242 KiB  
Article
Positive Solutions of a Fractional Thermostat Model with a Parameter
by Xinan Hao and Luyao Zhang
Symmetry 2019, 11(1), 122; https://doi.org/10.3390/sym11010122 - 21 Jan 2019
Cited by 14 | Viewed by 3002
Abstract
We study the existence, multiplicity, and uniqueness results of positive solutions for a fractional thermostat model. Our approach depends on the fixed point index theory, iterative method, and nonsymmetry property of the Green function. The properties of positive solutions depending on a parameter [...] Read more.
We study the existence, multiplicity, and uniqueness results of positive solutions for a fractional thermostat model. Our approach depends on the fixed point index theory, iterative method, and nonsymmetry property of the Green function. The properties of positive solutions depending on a parameter are also discussed. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)
15 pages, 267 KiB  
Article
Oscillatory Behavior of Three Dimensional α-Fractional Delay Differential Systems
by Adem Kilicman, Vadivel Sadhasivam, Muthusamy Deepa and Nagamanickam Nagajothi
Symmetry 2018, 10(12), 769; https://doi.org/10.3390/sym10120769 - 18 Dec 2018
Cited by 5 | Viewed by 2829
Abstract
In the present work we study the oscillatory behavior of three dimensional α -fractional nonlinear delay differential system. We establish some sufficient conditions that will ensure all solutions are either oscillatory or converges to zero, by using the inequality technique and generalized Riccati [...] Read more.
In the present work we study the oscillatory behavior of three dimensional α -fractional nonlinear delay differential system. We establish some sufficient conditions that will ensure all solutions are either oscillatory or converges to zero, by using the inequality technique and generalized Riccati transformation. The newly derived criterion are also used to establish a new class of systems with delay which are not covered in the existing study of literature. Further, we constructed some suitable illustrations. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)
13 pages, 257 KiB  
Article
Generalized Preinvex Functions and Their Applications
by Adem Kiliçman and Wedad Saleh
Symmetry 2018, 10(10), 493; https://doi.org/10.3390/sym10100493 - 13 Oct 2018
Cited by 11 | Viewed by 2664
Abstract
A class of function called sub-b-s-preinvex function is defined as a generalization of s-convex and b-preinvex functions, and some of its basic properties are presented here. The sufficient conditions of optimality for unconstrainded and inquality constrained programming are discussed under the sub-b-s-preinvexity. Moreover, [...] Read more.
A class of function called sub-b-s-preinvex function is defined as a generalization of s-convex and b-preinvex functions, and some of its basic properties are presented here. The sufficient conditions of optimality for unconstrainded and inquality constrained programming are discussed under the sub-b-s-preinvexity. Moreover, some new inequalities of the Hermite—Hadamard type for differentiable sub-b-s-preinvex functions are presented. Examples of applications of these inequalities are shown. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)
20 pages, 314 KiB  
Article
Global Mittag—Leffler Synchronization for Neural Networks Modeled by Impulsive Caputo Fractional Differential Equations with Distributed Delays
by Ravi Agarwal, Snezhana Hristova and Donal O’Regan
Symmetry 2018, 10(10), 473; https://doi.org/10.3390/sym10100473 - 10 Oct 2018
Cited by 12 | Viewed by 2304
Abstract
The synchronization problem for impulsive fractional-order neural networks with both time-varying bounded and distributed delays is studied. We study the case when the neural networks and the fractional derivatives of all neurons depend significantly on the moments of impulses and we consider both [...] Read more.
The synchronization problem for impulsive fractional-order neural networks with both time-varying bounded and distributed delays is studied. We study the case when the neural networks and the fractional derivatives of all neurons depend significantly on the moments of impulses and we consider both the cases of state coupling controllers and output coupling controllers. The fractional generalization of the Razumikhin method and Lyapunov functions is applied. Initially, a brief overview of the basic fractional derivatives of Lyapunov functions used in the literature is given. Some sufficient conditions are derived to realize the global Mittag–Leffler synchronization of impulsive fractional-order neural networks. Our results are illustrated with examples. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)
21 pages, 390 KiB  
Article
Multiplicity of Small or Large Energy Solutions for Kirchhoff–Schrödinger-Type Equations Involving the Fractional p-Laplacian in ℝN
by Jae-Myoung Kim, Yun-Ho Kim and Jongrak Lee
Symmetry 2018, 10(10), 436; https://doi.org/10.3390/sym10100436 - 26 Sep 2018
Cited by 4 | Viewed by 2494
Abstract
We herein discuss the following elliptic equations: [...] Read more.
We herein discuss the following elliptic equations: M R N R N | u ( x ) u ( y ) | p | x y | N + p s d x d y ( Δ ) p s u + V ( x ) | u | p 2 u = λ f ( x , u ) in R N , where ( Δ ) p s is the fractional p-Laplacian defined by ( Δ ) p s u ( x ) = 2 lim ε 0 R N \ B ε ( x ) | u ( x ) u ( y ) | p 2 ( u ( x ) u ( y ) ) | x y | N + p s d y , x R N . Here, B ε ( x ) : = { y R N : | x y | < ε } , V : R N ( 0 , ) is a continuous function and f : R N × R R is the Carathéodory function. Furthermore, M : R 0 + R + is a Kirchhoff-type function. This study has two aims. One is to study the existence of infinitely many large energy solutions for the above problem via the variational methods. In addition, a major point is to obtain the multiplicity results of the weak solutions for our problem under various assumptions on the Kirchhoff function M and the nonlinear term f. The other is to prove the existence of small energy solutions for our problem, in that the sequence of solutions converges to 0 in the L -norm. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)
15 pages, 296 KiB  
Article
Positive Solutions for a Three-Point Boundary Value Problem of Fractional Q-Difference Equations
by Chen Yang
Symmetry 2018, 10(9), 358; https://doi.org/10.3390/sym10090358 - 23 Aug 2018
Cited by 12 | Viewed by 2334
Abstract
In this work, a three-point boundary value problem of fractional q-difference equations is discussed. By using fixed point theorems on mixed monotone operators, some sufficient conditions that guarantee the existence and uniqueness of positive solutions are given. In addition, an iterative scheme [...] Read more.
In this work, a three-point boundary value problem of fractional q-difference equations is discussed. By using fixed point theorems on mixed monotone operators, some sufficient conditions that guarantee the existence and uniqueness of positive solutions are given. In addition, an iterative scheme can be made to approximate the unique solution. Finally, some interesting examples are provided to illustrate the main results. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory, Methods and Applications)
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