Symmetry and Solutions of Fractional Differential Equations with Their Developments

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

Deadline for manuscript submissions: 15 October 2024 | Viewed by 12199

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School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan 030031, China
Interests: methods and application of nonlinear equations; fractional calculus and their applications; boundary value problems; ordinary & partial differential equations; fractional differential equations; fractional Laplacian problem; analytical and numerical methods for nonlinear problems; methods of functional analysis; iteration methods for differential equations; Hessian equation; Monge–Ampere equation; modern analytical methods and their applications

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Guest Editor
Emeritus Research Professor of Department of Mathematics and Systems Engineering, Florida Institute of Technology, Melbourne, FL 32901, USA
Interests: nonlinear analysis; differential and difference equations; fixed point theory; general inequalities
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Guest Editor
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
Interests: boundary value problems; ordinary & partial differential equations; fractional differential equations; analytical and numerical methods for nonlinear problems; methods of functional analysis; stability theory; applications in energy problems; ecology; fluid mechanics; acoustic scattering; disease models
Special Issues, Collections and Topics in MDPI journals

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Guest Editor
School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan 030031, China
Interests: fractional calculus and applications; differential equations & nonlinear analysis; integral equation and inequalities; fractional Laplacian problem; Hessian equation; Monge–Ampere equation; modern analytical methods and their applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional differential equations arise in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics, fitting of experimental data, etc. In addition, it is very important to study the properties of the solutions of fractional differential equations, such as existence, uniqueness, symmetry, and monotonicity of the solutions. On the other hand, nonlinear functional analysis is an important branch of modern analytical mathematics, and it is one of the most active research areas in analytical mathematics. This Special Issue focuses on applying the tools of nonlinear functional analysis to study fractional differential equations and systems, in particular, fractional differential and integral operators, fractional Laplacian operators and their variants, fractional Hammerstein equations, fractional stochastic differential equations, etc. We welcome excellent manuscripts on a wide variety of nonlinear problems based on nonlinear analytical methods, including but not limited to cone theory, topological degree methods, upper and lower solution methods, critical point theory, monotonic iterative methods, and fixed-point methods, etc.

Prof. Lihong Zhang
Prof. Dr. Ravi P. Agarwal
Prof. Dr. Bashir Ahmad
Prof. Dr. Guotao Wang
Guest Editors

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Keywords

  • nonlinear fractional ordinary (partial) differential equations and applications
  • nonlinear problem involving fractional Laplacian operators and their variants
  • nonlinear Hammerstein equations involving fractional operators
  • nonlocal Monge–Ampere equation and its extensions
  • nonlocal operators, symmetries, and applications
  • geometrical methods for problems of mathematical physics
  • geometrical analysis and differential equations
  • monotonicity of solutions to various differential equations
  • existence and uniqueness of solutions to differential equations
  • symmetry of solutions to differential equations

Published Papers (9 papers)

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Research

16 pages, 301 KiB  
Article
On Higher-Order Nonlinear Fractional Elastic Equations with Dependence on Lower Order Derivatives in Nonlinearity
by Yujun Cui, Chunyu Liang and Yumei Zou
Fractal Fract. 2024, 8(7), 398; https://doi.org/10.3390/fractalfract8070398 - 2 Jul 2024
Viewed by 262
Abstract
The paper studied high-order nonlinear fractional elastic equations that depend on low-order derivatives in nonlinearity and established the existence and uniqueness results by using the Leray–Schauder alternative theorem and Perov’s fixed point theorem on an appropriate space under mild assumptions. Examples are given [...] Read more.
The paper studied high-order nonlinear fractional elastic equations that depend on low-order derivatives in nonlinearity and established the existence and uniqueness results by using the Leray–Schauder alternative theorem and Perov’s fixed point theorem on an appropriate space under mild assumptions. Examples are given to illustrate the key results. Full article
21 pages, 369 KiB  
Article
Study on a Nonlocal Fractional Coupled System Involving (k,ψ)-Hilfer Derivatives and (k,ψ)-Riemann–Liouville Integral Operators
by Ayub Samadi, Sotiris K. Ntouyas and Jessada Tariboon
Fractal Fract. 2024, 8(4), 211; https://doi.org/10.3390/fractalfract8040211 - 4 Apr 2024
Viewed by 785
Abstract
This paper deals with a nonlocal fractional coupled system of (k,ψ)-Hilfer fractional differential equations, which involve, in boundary conditions, (k,ψ)-Hilfer fractional derivatives and (k,ψ)-Riemann–Liouville fractional integrals. The existence [...] Read more.
This paper deals with a nonlocal fractional coupled system of (k,ψ)-Hilfer fractional differential equations, which involve, in boundary conditions, (k,ψ)-Hilfer fractional derivatives and (k,ψ)-Riemann–Liouville fractional integrals. The existence and uniqueness of solutions are established for the considered coupled system by using standard tools from fixed point theory. More precisely, Banach and Krasnosel’skiĭ’s fixed-point theorems are used, along with Leray–Schauder alternative. The obtained results are illustrated by constructed numerical examples. Full article
30 pages, 402 KiB  
Article
Existence of Solutions for the Initial Value Problem with Hadamard Fractional Derivatives in Locally Convex Spaces
by Weiwei Liu and Lishan Liu
Fractal Fract. 2024, 8(4), 191; https://doi.org/10.3390/fractalfract8040191 - 27 Mar 2024
Viewed by 819
Abstract
In this paper, we investigate an initial value problem for a nonlinear fractional differential equation on an infinite interval. The differential operator is taken in the Hadamard sense and the nonlinear term involves two lower-order fractional derivatives of the unknown function. In order [...] Read more.
In this paper, we investigate an initial value problem for a nonlinear fractional differential equation on an infinite interval. The differential operator is taken in the Hadamard sense and the nonlinear term involves two lower-order fractional derivatives of the unknown function. In order to establish the global existence criteria, we first verify that there exists a unique positive solution to an integral equation based on a class of new integral inequality. Next, we construct a locally convex space, which is metrizable and complete. On this space, applying Schäuder’s fixed point theorem, we obtain the existence of at least one solution to the initial value problem. Full article
16 pages, 1292 KiB  
Article
A Novel Hybrid Crossover Dynamics of Monkeypox Disease Mathematical Model with Time Delay: Numerical Treatments
by Nasser H. Sweilam, Seham M. Al-Mekhlafi, Saleh M. Hassan, Nehaya R. Alsenaideh and Abdelaziz E. Radwan
Fractal Fract. 2024, 8(4), 185; https://doi.org/10.3390/fractalfract8040185 - 24 Mar 2024
Viewed by 849
Abstract
In this paper, we improved a mathematical model of monkeypox disease with a time delay to a crossover model by incorporating variable-order and fractional differential equations, along with stochastic fractional derivatives, in three different time intervals. The stability and positivity of the solutions [...] Read more.
In this paper, we improved a mathematical model of monkeypox disease with a time delay to a crossover model by incorporating variable-order and fractional differential equations, along with stochastic fractional derivatives, in three different time intervals. The stability and positivity of the solutions for the proposed model are discussed. Two numerical methods are constructed to study the behavior of the proposed models. These methods are the nonstandard modified Euler Maruyama technique and the nonstandard Caputo proportional constant Adams-Bashfourth fifth step method. Many numerical experiments were conducted to verify the efficiency of the methods and support the theoretical results. This study’s originality is the use of fresh data simulation techniques and different solution methodologies. Full article
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23 pages, 437 KiB  
Article
Monotone Positive Radial Solution of Double Index Logarithm Parabolic Equations
by Mengru Liu and Lihong Zhang
Fractal Fract. 2024, 8(3), 173; https://doi.org/10.3390/fractalfract8030173 - 16 Mar 2024
Viewed by 974
Abstract
This article mainly studies the double index logarithmic nonlinear fractional g-Laplacian parabolic equations with the Marchaud fractional time derivatives tα. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double [...] Read more.
This article mainly studies the double index logarithmic nonlinear fractional g-Laplacian parabolic equations with the Marchaud fractional time derivatives tα. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double non-locality of space-time and the nonlinearity of the fractional g-Laplacian, we establish the unbounded narrow domain principle, which provides a starting point for the moving plane method. Meanwhile, for the purpose of eliminating the assumptions of boundedness on the solutions, the averaging effects of a non-local operator are established; then, these averaging effects are applied twice to ensure that the plane can be continuously moved toward infinity. Based on the above, the monotonicity of a positive solution for the above fractional g-Laplacian parabolic equations is studied. Full article
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19 pages, 342 KiB  
Article
On a Faster Iterative Method for Solving Fractional Delay Differential Equations in Banach Spaces
by James Abah Ugboh, Joseph Oboyi, Mfon Okon Udo, Hossam A. Nabwey, Austine Efut Ofem and Ojen Kumar Narain
Fractal Fract. 2024, 8(3), 166; https://doi.org/10.3390/fractalfract8030166 - 14 Mar 2024
Viewed by 930
Abstract
In this paper, we consider a faster iterative method for approximating the fixed points of generalized α-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our [...] Read more.
In this paper, we consider a faster iterative method for approximating the fixed points of generalized α-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our findings, we present some nontrivial examples of the considered mappings. Furthermore, we show that the class of mappings considered is more general than some nonexpansive-type mappings. Also, we show numerically that the method studied in our article is more efficient than several existing methods. Lastly, we use our main results to approximate the solution of a delay fractional differential equation in the Caputo sense. Our results generalize and improve many well-known existing results. Full article
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33 pages, 1380 KiB  
Article
Proposal for Use of the Fractional Derivative of Radial Functions in Interpolation Problems
by Anthony Torres-Hernandez, Fernando Brambila-Paz and Rafael Ramirez-Melendez
Fractal Fract. 2024, 8(1), 16; https://doi.org/10.3390/fractalfract8010016 - 23 Dec 2023
Viewed by 4032
Abstract
This paper presents the construction of a family of radial functions aimed at emulating the behavior of the radial basis function known as thin plate spline (TPS). Additionally, a method is proposed for applying fractional derivatives, both partially and fully, to these functions [...] Read more.
This paper presents the construction of a family of radial functions aimed at emulating the behavior of the radial basis function known as thin plate spline (TPS). Additionally, a method is proposed for applying fractional derivatives, both partially and fully, to these functions for use in interpolation problems. Furthermore, a technique is employed to precondition the matrices generated in the presented problems through QR decomposition. Similarly, a method is introduced to define two different types of abelian groups for any fractional operator defined in the interval [0,1), among which the Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative, and Caputo fractional derivative are worth mentioning. Finally, a form of radial interpolant is suggested for application in solving fractional differential equations using the asymmetric collocation method, and examples of its implementation in differential operators utilizing the aforementioned fractional operators are shown. Full article
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24 pages, 724 KiB  
Article
Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications
by Junjie Wang, Shoucheng Yuan and Xiao Liu
Fractal Fract. 2023, 7(12), 868; https://doi.org/10.3390/fractalfract7120868 - 6 Dec 2023
Cited by 1 | Viewed by 1082
Abstract
The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and [...] Read more.
The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes. Full article
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14 pages, 312 KiB  
Article
Maximum Principle for Variable-Order Fractional Conformable Differential Equation with a Generalized Tempered Fractional Laplace Operator
by Tingting Guan and Lihong Zhang
Fractal Fract. 2023, 7(11), 798; https://doi.org/10.3390/fractalfract7110798 - 1 Nov 2023
Viewed by 1226
Abstract
In this paper, we investigate properties of solutions to a space-time fractional variable-order conformable nonlinear differential equation with a generalized tempered fractional Laplace operatorby using the maximum principle. We first establish some new important fractional various-order conformable inequalities. With these inequalities, we prove [...] Read more.
In this paper, we investigate properties of solutions to a space-time fractional variable-order conformable nonlinear differential equation with a generalized tempered fractional Laplace operatorby using the maximum principle. We first establish some new important fractional various-order conformable inequalities. With these inequalities, we prove a new maximum principle with space-time fractional variable-order conformable derivatives and a generalized tempered fractional Laplace operator. Moreover, we discuss some results about comparison principles and properties of solutions for a family of space-time fractional variable-order conformable nonlinear differential equations with a generalized tempered fractional Laplace operator by maximum principle. Full article
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