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Fractional Differential Equations in Anomalous Diffusion

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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 (31 December 2022) | Viewed by 17940

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Special Issue Editors

Dr. Xinguang Zhang

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1. School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China
2. Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia
Interests: nonlinear analysis on manifolds; fractional order differential equations; partial differential equation; variational methods; fixed points theorem; critical points theory; singular nonlinear systems; fractional calculus; mathematical modeling
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Yonghong Wu

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Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia
Interests: computational mathematics; applied mathematical modelling; differential equations and boundary value problems; fluid mechanics
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Prof. Dr. Chuanjun Chen

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Guest Editor

School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China
Interests: computational mathematics; numerical method for partial differential equations; phase-field models
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Dr. Jiwei Zhang

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Guest Editor

School of Computer Science (National Pilot Software Engineering School), Beijing University of Posts and Telecommunications, Beijing 100876, China
Interests: computer vision; information security
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Dr. Chun Lu

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Guest Editor

Department of Mathematics, Qingdao University of Technology, Qingdao 266520, China
Interests: stochastic differential equation; fractional order differential equations; asymptotic behavior of biological model; stability analysis of epidemic model; computational mathematics
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Special Issue Information

Dear Colleagues,

The phenomenon of anomalous diffusion is an important dynamic behavior arising from many applied science fields, such as turbulence, (see page on porous media and pollution control). Thus, nonlinear modelling is a better strategy to depict a variety of complex anomalous diffusion phenomena. However, modelling anomalous diffusion using the differential equation is still a perplexing mathematical physics issue. Because the fractal or the fractional derivative can well describe the inherent abnormal-exponential or heavy-tail decay processes, in recent decades fractal and fractional derivatives have been used to model many anomalous diffusion processes.

Thus, fractional differential equation in anomalous diffusion has become a new research area of analytical mathematics, providing useful tools to model many problems arising from mathematical physics, fluid dynamics, chemistry, biology, economics, control theory and image processing with memory effects.

We invite researchers to submit original research as well as review articles discussing the recent development of the fractional differential equation in anomalous diffusion and its applications in sciences, technologies and engineering.

Topics include (but are not limited to):

Theory of the fractal or the fractional derivative.
Initial and boundary value problems of fractional differential equations in anomalous diffusion.
Inequalities of fractional integrals and fractional derivatives.
Singular and impulsive fractional differential and integral equations.
Analysis and control in the fractal or the fractional differential equations with anomalous diffusion.
Numerical analysis and algorithm for fractional differential equations.
Fixed point theory and application in fractional calculus.
Fractional functional equations in function spaces.
Fractional network arising in physical models.
Fractional stochastic differential equations.

Dr. Xinguang Zhang
Prof. Dr. Yonghong Wu
Prof. Dr. Chuanjun Chen
Dr. Jiwei Zhang
Dr. Chun Lu
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

fractal or the fractional derivative
initial and boundary value problems
fractional differential equations
anomalous diffusion
fractional network
numerical analysis and algorithm

Published Papers (12 papers)

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Research

25 pages, 592 KiB

Open AccessArticle

Dynamic Properties for a Second-Order Stochastic SEIR Model with Infectivity in Incubation Period and Homestead-Isolation of the Susceptible Population

by Chun Lu, Honghui Liu and Junhua Zhou

Fractal Fract. 2023, 7(5), 365; https://doi.org/10.3390/fractalfract7050365 - 28 Apr 2023

Viewed by 1930

Abstract

In this article, we analyze a second-order stochastic SEIR epidemic model with latent infectious and susceptible populations isolated at home. Firstly, by putting forward a novel inequality, we provide a criterion for the presence of an ergodic stationary distribution of the model. Secondly, we establish sufficient conditions for extinction. Thirdly, by solving the corresponding Fokker–Plank equation, we derive the probability density function around the quasi-endemic equilibrium of the stochastic model. Finally, by using the epidemic data of the corresponding deterministic model, two numerical tests are presented to illustrate the validity of the theoretical results. Our conclusions demonstrate that nations should persevere in their quarantine policies to curb viral transmission when the COVID-19 pandemic proceeds to spread internationally. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

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13 pages, 3565 KiB

Open AccessArticle

Dynamics and Stability of a Fractional-Order Tumor–Immune Interaction Model with B-D Functional Response and Immunotherapy

by Xiaozhou Feng, Mengyan Liu, Yaolin Jiang and Dongping Li

Fractal Fract. 2023, 7(2), 200; https://doi.org/10.3390/fractalfract7020200 - 17 Feb 2023

Cited by 1 | Viewed by 1118

Abstract

In this paper, we investigate a fractional-order tumor–immune interaction model with B-D function item and immunotherapy. First, the existence, uniqueness and nonnegativity of the solutions of the model are established. Second, the local and global asymptotic stability of some tumor-free equilibrium points and a unique positive equilibrium point are obtained. Finally, we use numerical simulation method to visualize and verify the theoretical conclusions. It is known that the fractional-order parameter

β

has a stabilization effect, and the tumor cells can be destroyed or controlled by using immunotherapy. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

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19 pages, 5218 KiB

Open AccessArticle

A Compact Scheme Combining the Fast Time Stepping Method for Solving 2D Fractional Subdiffusion Equations

by Yibin Xu, Yanqin Liu, Xiuling Yin, Libo Feng and Zihua Wang

Fractal Fract. 2023, 7(2), 186; https://doi.org/10.3390/fractalfract7020186 - 13 Feb 2023

Viewed by 1408

Abstract

In this paper, in order to improve the calculation accuracy and efficiency of α-order Caputo fractional derivative (0 < α ≤ 1), we developed a compact scheme combining the fast time stepping method for solving 2D fractional nonlinear subdiffusion equations. In the temporal direction, a time stepping method was applied. It can reach second-order accuracy. In the spatial direction, we utilized the compact difference scheme, which can reach fourth-order accuracy. Some properties of coefficients are given, which are essential for the theoretical analysis. Meanwhile, we rigorously proved the unconditional stability of the proposed scheme and gave the sharp error estimate. To overcome the intensive computation caused by the fractional operators, we combined a fast algorithm, which can reduce the computational complexity from O(N²) to O(Nlog(N)), where N represents the number of time steps. Considering that the solution of the subdiffusion equation is weakly regular in most cases, we added correction terms to ensure that the solution can achieve the optimal convergence accuracy. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

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19 pages, 633 KiB

Open AccessArticle

A Class of Fractional Stochastic Differential Equations with a Soft Wall

by Kęstutis Kubilius and Aidas Medžiūnas

Fractal Fract. 2023, 7(2), 110; https://doi.org/10.3390/fractalfract7020110 - 21 Jan 2023

Cited by 1 | Viewed by 1065

Abstract

In this paper we are interested in fractional stochactic differential equations (SDEs) with a soft wall. What do we mean by such a type of equation? It has been established that SDE with reflection can be imagined as equations having a hard wall. Now, by introducing repulsion instead of reflection, one obtains an SDE with a soft wall. In contrast to the SDE with reflection, where the process cannot pass the hard wall, the soft wall is repulsive but not impenetrable. As the process crosses the soft wall boundary, it experiences the force of a chosen magnitude in the opposite direction. When the process is far from the wall, the force acts weakly. We find conditions under which SDE with a soft wall has a unique solution and construct an implicit Euler approximation with a rate of convergence for this equation. Using the example of the fractional Vasicek process with soft walls, we illustrate the dependence of the behaviour of the solution on the repulsion force. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

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19 pages, 6885 KiB

Open AccessArticle

Conservative Continuous-Stage Stochastic Runge–Kutta Methods for Stochastic Differential Equations

by Xiuyan Li, Zhenyu Wang, Qiang Ma and Xiaohua Ding

Fractal Fract. 2023, 7(1), 83; https://doi.org/10.3390/fractalfract7010083 - 11 Jan 2023

Viewed by 1298

Abstract

In this paper, we develop a new class of conservative continuous-stage stochastic Runge–Kutta methods for solving stochastic differential equations with a conserved quantity. The order conditions of the continuous-stage stochastic Runge–Kutta methods are given based on the theory of stochastic B-series and multicolored rooted tree. Sufficient conditions for the continuous-stage stochastic Runge–Kutta methods preserving the conserved quantity of stochastic differential equations are derived in terms of the coefficients. Conservative continuous-stage stochastic Runge–Kutta methods of mean square convergence order 1 for general stochastic differential equations, as well as conservative continuous-stage stochastic Runge–Kutta methods of high order for single integrand stochastic differential equations, are constructed. Numerical experiments are performed to verify the conservative property and the accuracy of the proposed methods in the longtime simulation. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

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16 pages, 358 KiB

Open AccessArticle

Threshold Results for the Existence of Global and Blow-Up Solutions to a Time Fractional Diffusion System with a Nonlinear Memory Term in a Bounded Domain

by Quanguo Zhang and Yaning Li

Fractal Fract. 2023, 7(1), 56; https://doi.org/10.3390/fractalfract7010056 - 2 Jan 2023

Cited by 1 | Viewed by 1032

Abstract

In this paper, we consider a time fractional diffusion system with a nonlinear memory term in a bounded domain. We mainly prove some blow-up and global existence results for this problem. Moreover, we also give the decay estimates of the global solutions. Our proof relies on the eigenfunction method combined with the asymptotic behavior of the solution of a fractional differential inequality system, the estimates of the solution operators and the asymptotic behavior of the Mittag–Leffler function. In particular, we give the critical exponents of this problem in different cases. Our results show that, in some cases, whether one of the initial values is identically equal to zero has a great influence on blow-up and global existence of the solutions for this problem, which is a remarkable property of time fractional diffusion systems because the classical diffusion systems can not admit this property. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

24 pages, 386 KiB

Open AccessArticle

Solvability of Nonlinear Impulsive Generalized Fractional Differential Equations with (p,q)-Laplacian Operator via Critical Point Theory

by Jianwen Zhou, Yuqiong Liu, Yanning Wang and Jianfeng Suo

Fractal Fract. 2022, 6(12), 719; https://doi.org/10.3390/fractalfract6120719 - 2 Dec 2022

Cited by 1 | Viewed by 1184

Abstract

In this paper, we consider the nonlinear impulsive generalized fractional differential equations with

(p, q)

-Laplacian operator for

1 < p \leq q < \infty

, in which the nonlinearity f contains two fractional derivatives with respect to another function. [...] Read more.

In this paper, we consider the nonlinear impulsive generalized fractional differential equations with

(p, q)

-Laplacian operator for

1 < p \leq q < \infty

, in which the nonlinearity f contains two fractional derivatives with respect to another function. Since the complexity of the nonlinear term and the impulses exist in generalized fractional calculus, it is difficult to find the corresponding variational functional of the problem. The existence of nontrivial solutions for the problem is established by the mountain pass theorem and iterative technique under some appropriate assumptions. Furthermore, our main result is demonstrated by an illustrative example to show its feasibility and effectiveness. Due to the employment of a generalized fractional operator, our results extend some existing research findings. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

28 pages, 460 KiB

Open AccessArticle

Properties of Hadamard Fractional Integral and Its Application

by Weiwei Liu and Lishan Liu

Fractal Fract. 2022, 6(11), 670; https://doi.org/10.3390/fractalfract6110670 - 13 Nov 2022

Cited by 2 | Viewed by 1399

Abstract

We begin by introducing some function spaces

L_{c}^{p} (R^{+}), X_{c}^{p} (J)

made up of integrable functions with exponent or power weights defined on infinite intervals, and then we investigate the properties of Mellin [...] Read more.

We begin by introducing some function spaces

L_{c}^{p} (R^{+}), X_{c}^{p} (J)

made up of integrable functions with exponent or power weights defined on infinite intervals, and then we investigate the properties of Mellin convolution operators mapping on these spaces, next, we derive some new boundedness and continuity properties of Hadamard integral operators mapping on

X_{c}^{p} (J)

and

X^{p} (J)

. Based on this, we investigate a class of boundary value problems for Hadamard fractional differential equations with the integral boundary conditions and the disturbance parameters, and obtain uniqueness results for positive solutions to the boundary value problem under some weaker conditions. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

13 pages, 415 KiB

Open AccessArticle

A Second-Order Adaptive Grid Method for a Singularly Perturbed Volterra Integrodifferential Equation

by Libin Liu, Ying Liang and Yong Zhang

Fractal Fract. 2022, 6(11), 636; https://doi.org/10.3390/fractalfract6110636 - 1 Nov 2022

Viewed by 1112

Abstract

In this paper, an adaptive grid method for a singularly perturbed Volterra integro-differential equation is studied. Firstly, this problem is discretized by a new second-order finite difference scheme, for which a truncation error analysis is conducted. Then, based on this truncation error bound and the mesh equidistribution principle, we show that there is a mesh that provides an optimal error bound of O(N⁻²), which is robust with respect to the perturbation parameter. Finally, based on an approximation monitor function, an adaptive grid generation algorithm is constructed and some numerical results are given to support our theoretical results. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

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18 pages, 327 KiB

Open AccessArticle

Positive Solutions for a System of Riemann–Liouville Type Fractional-Order Integral Boundary Value Problems

by Keyu Zhang, Fehaid Salem Alshammari, Jiafa Xu and Donal O’Regan

Fractal Fract. 2022, 6(9), 480; https://doi.org/10.3390/fractalfract6090480 - 29 Aug 2022

Cited by 1 | Viewed by 1101

Abstract

In this paper, we use the fixed-point index to establish positive solutions for a system of Riemann–Liouville type fractional-order integral boundary value problems. Some appropriate concave and convex functions are used to characterize coupling behaviors of our nonlinearities. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

14 pages, 331 KiB

Open AccessArticle

Existence of Positive Solutions for a Singular Second-Order Changing-Sign Differential Equation on Time Scales

by Hui Tian, Xinguang Zhang, Yonghong Wu and Benchawan Wiwatanapataphee

Fractal Fract. 2022, 6(6), 315; https://doi.org/10.3390/fractalfract6060315 - 3 Jun 2022

Cited by 6 | Viewed by 1392

Abstract

In this paper, we focus on the existence of positive solutions for a boundary value problem of the changing-sign differential equation on time scales. By constructing a translation transformation and combining with the properties of the solution of the nonhomogeneous boundary value problem, we transfer the changing-sign problem to a positone problem, then by means of the known fixed-point theorem, several sufficient conditions for the existence of positive solutions are established for the case in which the nonlinear term of the equation may change sign. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

24 pages, 1657 KiB

Open AccessArticle

A Visually Secure Image Encryption Based on the Fractional Lorenz System and Compressive Sensing

by Hua Ren, Shaozhang Niu, Jiajun Chen, Ming Li and Zhen Yue

Fractal Fract. 2022, 6(6), 302; https://doi.org/10.3390/fractalfract6060302 - 29 May 2022

Cited by 12 | Viewed by 2138

Abstract

Recently, generating visually secure cipher images by compressive sensing (CS) techniques has drawn much attention among researchers. However, most of these algorithms generate cipher images based on direct bit substitution and the underlying relationship between the hidden and modified data is not considered, which reduces the visual security of cipher images. In addition, performing CS on plain images directly is inefficient, and CS decryption quality is not high enough. Thus, we design a novel cryptosystem by introducing vector quantization (VQ) into CS-based encryption based on a 3D fractional Lorenz chaotic system. In our work, CS compresses only the sparser error matrix generated from the plain and VQ images in the secret generation phase, which improves CS compression performance and the quality of decrypted images. In addition, a smooth function is used in the embedding phase to find the underlying relationship and determine relatively suitable modifiable values for the carrier image. All the secret streams are produced by updating the initial values and control parameters from the fractional chaotic system, and then utilized in CS, diffusion, and embedding. Simulation results demonstrate the effectiveness of the proposed method. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

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