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Fractal Fract
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Recent Advances in Fractional Differential Equations and Their Applications, 2nd...
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Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Numerical and Computational Methods".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 12207

Special Issue Editors

Dr. Xiaoli Chen

E-Mail Website
Guest Editor
Department of Mathematics, National University of Singapore, Singapore, Singapore
Interests: fractional partial differential equation; machine learning; stochastic dynamical systems
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Dongfang Li

grade E-Mail Website
Guest Editor
Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China
Interests: numerical solutions to PDE and ODE; fractional partial differential equation
Special Issues, Collections and Topics in MDPI journals
Dr. Ting Gao

E-Mail Website
Guest Editor
School of Mathematics and Statistics and Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, China
Interests: machine learning; stochastic dynamical systems

Special Issue Information

Dear Colleagues,

Fractional differential equations describe the dynamic systems of complex and non-local systems with memory. They can be developed from stochastic dynamical systems driven by non-Gaussian Levy noise, which have long tails and bursting sample routes. They feature in a wide variety of scientific and engineering sectors, including physics, biology, economics, and chemical engineering. Due to memory and nonlocality issues, finding analytical solutions can be challenging, and identifying effective strategies for numerically solving fractional differential equations is a pressing issue.

Potential topics for this Special Issue include (but are not limited to):

  • New numerical methods for time fractional differential equations;
  • New numerical methods for space fractional (nonlocal) differential equations;
  • The relationship between stochastic differential equations and nonlocal differential equations;
  • Regularity estimate and homogenization for nonlocal differential equations;
  • Application of stochastic dynamics and fractional models;
  • Machine learning methods for FDEs
  • Inverse problems in non-local PDE / SDE
  • Effective dynamics and reduced order models

Dr. Xiaoli Chen
Prof. Dr. Dongfang Li
Dr. Ting Gao
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

  • nonlocal differential equation
  • fractional differential equation
  • stochastic differential equation
  • numerical method
  • machine learning methods

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Further information on MDPI's Special Issue polices can be found here.

Related Special Issue

Published Papers (12 papers)

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Research

13 pages, 324 KiB  
Article
φ−Hilfer Fractional Cauchy Problems with Almost Sectorial and Lie Bracket Operators in Banach Algebras
by Faten H. Damag, Amin Saif and Adem Kiliçman
Fractal Fract. 2024, 8(12), 741; https://doi.org/10.3390/fractalfract8120741 - 16 Dec 2024
Viewed by 446
Abstract
In the theory of Banach algebras, we use the Schauder fixed-point theorem to obtain some results that concern the existence property for mild solutions of fractional Cauchy problems that involve the Lie bracket operator, the almost sectorial operator, and the φ−Hilfer [...] Read more.
In the theory of Banach algebras, we use the Schauder fixed-point theorem to obtain some results that concern the existence property for mild solutions of fractional Cauchy problems that involve the Lie bracket operator, the almost sectorial operator, and the φ−Hilfer derivative operator. For any Banach algebra and in two types of non-compact associated semigroups and compact associated semigroups, we prove some properties of the existence of these mild solutions using the Hausdorff measure of a non-compact associated semigroup in the collection of bounded sets. That is, we obtain the existence property of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as non-compact. Some examples are introduced as applications for our results in commutative real Banach algebra R and commutative Banach algebra of the collection of continuous functions in R. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
9 pages, 678 KiB  
Article
A Note on Exact Results for Burgers-like Equations Involving Laguerre Derivatives
by Roberto Garra, Giuseppe Dattoli and Riccardo Droghei
Fractal Fract. 2024, 8(12), 723; https://doi.org/10.3390/fractalfract8120723 - 9 Dec 2024
Viewed by 474
Abstract
In this work, we consider some Burgers-like equations involving Laguerre derivatives and demonstrate that it is possible to construct specific exact solutions using separation of variables. We prove that a general scheme exists for constructing exact solutions for these Burgers-like equations and extending [...] Read more.
In this work, we consider some Burgers-like equations involving Laguerre derivatives and demonstrate that it is possible to construct specific exact solutions using separation of variables. We prove that a general scheme exists for constructing exact solutions for these Burgers-like equations and extending to more general cases, including nonlinear time-fractional equations. Exact solutions can also be obtained for KDV-like equations involving Laguerre derivatives. We finally consider a particular class of Burgers equations with variable coefficients whose solutions can be obtained similarly. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
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22 pages, 1032 KiB  
Article
Properties and Applications of Complex Fractal–Fractional Operators in the Open Unit Disk
by Adel A. Attiya, Soheil Salahshour, Rabha W. Ibrahim and Mansour F. Yassen
Fractal Fract. 2024, 8(10), 584; https://doi.org/10.3390/fractalfract8100584 - 3 Oct 2024
Viewed by 611
Abstract
A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function: Γℓ(.)). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows [...] Read more.
A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function: Γℓ(.)). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows the normalization associated with every operator. We investigate a number of significant geometric features thanks to this. Additionally, some integrals, such the Alexander and Libra integral operators, are associated with these operators. Simple power functions are among the illustrations that are provided. Additionally, the formulation of the discrete ℓ−fractal–fractional operators is conducted. We demonstrate that well-known examples are involved in the extended operators. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
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14 pages, 1503 KiB  
Article
Modelling Yeast Prion Dynamics: A Fractional Order Approach with Predictor–Corrector Algorithm
by Daasara Keshavamurthy Archana, Doddabhadrappla Gowda Prakasha and Nasser Bin Turki
Fractal Fract. 2024, 8(9), 542; https://doi.org/10.3390/fractalfract8090542 - 19 Sep 2024
Viewed by 559
Abstract
This work aims to comprehend the dynamics of neurodegenerative disease using a mathematical model of fractional-order yeast prions. In the context of the Caputo fractional derivative, we here study and examine the solution of this model using the Predictor–Corrector approach. An analysis has [...] Read more.
This work aims to comprehend the dynamics of neurodegenerative disease using a mathematical model of fractional-order yeast prions. In the context of the Caputo fractional derivative, we here study and examine the solution of this model using the Predictor–Corrector approach. An analysis has been conducted on the existence and uniqueness of the selected model. Also, we examined the model’s stability and the existence of equilibrium points. With the purpose of analyzing the dynamics of the Sup35 monomer and Sup35 prion population, we displayed the graphs to show the obtained solutions over time. Graphical simulations show that the behaviour of the populations can change based on fractional orders and threshold parameter values. This work may present a good example of how biological theories and data can be better understood via mathematical modelling. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
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12 pages, 325 KiB  
Article
Lp(Lq)-Maximal Regularity for Damped Equations in a Cylindrical Domain
by Edgardo Alvarez, Stiven Díaz and Carlos Lizama
Fractal Fract. 2024, 8(9), 516; https://doi.org/10.3390/fractalfract8090516 - 30 Aug 2024
Viewed by 886
Abstract
We show maximal regularity estimates for the damped hyperbolic and strongly damped wave equations with periodic initial conditions in a cylindrical domain. We prove that this property strongly depends on a critical combination on the parameters of the equation. Noteworthy, our results are [...] Read more.
We show maximal regularity estimates for the damped hyperbolic and strongly damped wave equations with periodic initial conditions in a cylindrical domain. We prove that this property strongly depends on a critical combination on the parameters of the equation. Noteworthy, our results are still valid for fractional powers of the negative Laplacian operator. We base our methods on the theory of operator-valued Fourier multipliers on vector-valued Lebesgue spaces of periodic functions. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
26 pages, 657 KiB  
Article
Spectral Galerkin Methods for Riesz Space-Fractional Convection–Diffusion Equations
by Xinxia Zhang, Jihan Wang, Zhongshu Wu, Zheyi Tang and Xiaoyan Zeng
Fractal Fract. 2024, 8(7), 431; https://doi.org/10.3390/fractalfract8070431 - 22 Jul 2024
Cited by 1 | Viewed by 855
Abstract
This paper applies the spectral Galerkin method to numerically solve Riesz space-fractional convection–diffusion equations. Firstly, spectral Galerkin algorithms were developed for one-dimensional Riesz space-fractional convection–diffusion equations. The equations were solved by discretizing in space using the Galerkin–Legendre spectral approaches and in time using [...] Read more.
This paper applies the spectral Galerkin method to numerically solve Riesz space-fractional convection–diffusion equations. Firstly, spectral Galerkin algorithms were developed for one-dimensional Riesz space-fractional convection–diffusion equations. The equations were solved by discretizing in space using the Galerkin–Legendre spectral approaches and in time using the Crank–Nicolson Leap-Frog (CNLF) scheme. In addition, the stability and convergence of semi-discrete and fully discrete schemes were analyzed. Secondly, we established a fully discrete form for the two-dimensional case with an additional complementary term on the left and then obtained the stability and convergence results for it. Finally, numerical simulations were performed, and the results demonstrate the effectiveness of our numerical methods. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
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25 pages, 3439 KiB  
Article
Split-Step Galerkin FE Method for Two-Dimensional Space-Fractional CNLS
by Xiaogang Zhu, Yaping Zhang and Yufeng Nie
Fractal Fract. 2024, 8(7), 402; https://doi.org/10.3390/fractalfract8070402 - 5 Jul 2024
Viewed by 1055
Abstract
In this paper, we study a split-step Galerkin finite element (FE) method for the two-dimensional Riesz space-fractional coupled nonlinear Schrödinger equations (CNLSs). The proposed method adopts a second-order split-step technique to handle the nonlinearity and FE approximation to discretize the fractional derivatives in [...] Read more.
In this paper, we study a split-step Galerkin finite element (FE) method for the two-dimensional Riesz space-fractional coupled nonlinear Schrödinger equations (CNLSs). The proposed method adopts a second-order split-step technique to handle the nonlinearity and FE approximation to discretize the fractional derivatives in space, which avoids iteration at each time layer. The analysis of mass conservative and convergent properties for this split-step FE scheme is performed. To test its capability, some numerical tests and the simulation of the double solitons intersection and plane wave are carried out. The results and comparisons with the algorithm combined with Newton’s iteration illustrate its effectiveness and advantages in computational efficiency. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
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10 pages, 290 KiB  
Article
Time-Stepping Error Estimates of Linearized Grünwald–Letnikov Difference Schemes for Strongly Nonlinear Time-Fractional Parabolic Problems
by Hongyu Qin, Lili Li, Yuanyuan Li and Xiaoli Chen
Fractal Fract. 2024, 8(7), 390; https://doi.org/10.3390/fractalfract8070390 - 29 Jun 2024
Cited by 2 | Viewed by 864
Abstract
A fully discrete scheme is proposed for numerically solving the strongly nonlinear time-fractional parabolic problems. Time discretization is achieved by using the Grünwald–Letnikov (G–L) method and some linearized techniques, and spatial discretization is achieved by using the standard second-order central difference scheme. Through [...] Read more.
A fully discrete scheme is proposed for numerically solving the strongly nonlinear time-fractional parabolic problems. Time discretization is achieved by using the Grünwald–Letnikov (G–L) method and some linearized techniques, and spatial discretization is achieved by using the standard second-order central difference scheme. Through a Grönwall-type inequality and some complementary discrete kernels, the optimal time-stepping error estimates of the proposed scheme are obtained. Finally, several numerical examples are given to confirm the theoretical results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
10 pages, 972 KiB  
Article
Ultrafast Diffusion Modeling via the Riemann–Liouville Nonlocal Structural Derivative and Its Application in Porous Media
by Wei Xu, Hui Liu, Lijuan Chen and Yongtao Zhou
Fractal Fract. 2024, 8(2), 110; https://doi.org/10.3390/fractalfract8020110 - 12 Feb 2024
Viewed by 1325
Abstract
Ultrafast diffusion disperses faster than super-diffusion, and this has been proven by several theoretical and experimental investigations. The mean square displacement of ultrafast diffusion grows exponentially, which provides a significant challenge for modeling. Due to the inhomogeneity, nonlinear interactions, and high porosity of [...] Read more.
Ultrafast diffusion disperses faster than super-diffusion, and this has been proven by several theoretical and experimental investigations. The mean square displacement of ultrafast diffusion grows exponentially, which provides a significant challenge for modeling. Due to the inhomogeneity, nonlinear interactions, and high porosity of cement materials, the motion of particles on their surfaces satisfies the conditions for ultrafast diffusion. The investigation of the diffusion behavior in cementitious materials is crucial for predicting the mechanical properties of cement. In this study, we first attempted to investigate the dynamic of ultrafast diffusion in cementitious materials underlying the Riemann–Liouville nonlocal structural derivative. We constructed a Riemann–Liouville nonlocal structural derivative ultrafast diffusion model with an exponential function and then extended the modeling strategy using the Mittag–Leffler function. The mean square displacement is analogous to the integral of the corresponding structural derivative, providing a reference standard for the selection of structural functions in practical applications. Based on experimental data on cement mortar, the accuracy of the Riemann–Liouville nonlocal structural derivative ultrafast diffusion model was verified. Compared to the power law diffusion and the exponential law diffusion, the mean square displacement with respect to the Mittag–Leffler law is closely tied to the actual data. The modeling approach based on the Riemann–Liouville nonlocal structural derivative provides an efficient tool for depicting ultrafast diffusion in porous media. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
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13 pages, 460 KiB  
Article
Error Analysis of the Nonuniform Alikhanov Scheme for the Fourth-Order Fractional Diffusion-Wave Equation
by Zihao An and Chaobao Huang
Fractal Fract. 2024, 8(2), 106; https://doi.org/10.3390/fractalfract8020106 - 10 Feb 2024
Cited by 1 | Viewed by 1301
Abstract
This paper considers the numerical approximation to the fourth-order fractional diffusion-wave equation. Using a separation of variables, we can construct the exact solution for such a problem and then analyze its regularity. The obtained regularity result indicates that the solution behaves as a [...] Read more.
This paper considers the numerical approximation to the fourth-order fractional diffusion-wave equation. Using a separation of variables, we can construct the exact solution for such a problem and then analyze its regularity. The obtained regularity result indicates that the solution behaves as a weak singularity at the initial time. Using the order reduction method, the fourth-order fractional diffusion-wave equation can be rewritten as a coupled system of low order, which is approximated by the nonuniform Alikhanov scheme in time and the finite difference method in space. Furthermore, the H2-norm stability result is obtained. With the help of this result and a priori bounds of the solution, an α-robust error estimate with optimal convergence order is derived. In order to further verify the accuracy of our theoretical analysis, some numerical results are provided. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
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17 pages, 417 KiB  
Article
Collocation-Based Approximation for a Time-Fractional Sub-Diffusion Model
by Kaido Lätt, Arvet Pedas, Hanna Britt Soots and Mikk Vikerpuur
Fractal Fract. 2023, 7(9), 657; https://doi.org/10.3390/fractalfract7090657 - 31 Aug 2023
Viewed by 969
Abstract
We consider the numerical solution of a one-dimensional time-fractional diffusion problem, where the order of the Caputo time derivative belongs to (0, 1). Using the technique of the method of lines, we first develop from the original problem a system of fractional ordinary [...] Read more.
We consider the numerical solution of a one-dimensional time-fractional diffusion problem, where the order of the Caputo time derivative belongs to (0, 1). Using the technique of the method of lines, we first develop from the original problem a system of fractional ordinary differential equations. Using an integral equation reformulation of this system, we study the regularity properties of the exact solution of the system of fractional differential equations and construct a piecewise polynomial collocation method to solve it numerically. We also investigate the convergence and the convergence order of the proposed method. To conclude, we present the results of some numerical experiments. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
10 pages, 296 KiB  
Article
Existence and Stability Results for Piecewise Caputo–Fabrizio Fractional Differential Equations with Mixed Delays
by Doha A. Kattan and Hasanen A. Hammad
Fractal Fract. 2023, 7(9), 644; https://doi.org/10.3390/fractalfract7090644 - 24 Aug 2023
Cited by 5 | Viewed by 1385
Abstract
In this article, by using the differential Caputo–Fabrizio operator, we suggest a novel family of piecewise differential equations (DEs). The issue under study contains a mixed delay period under the criteria of anti-periodic boundaries. It is possible to utilize the piecewise derivative to [...] Read more.
In this article, by using the differential Caputo–Fabrizio operator, we suggest a novel family of piecewise differential equations (DEs). The issue under study contains a mixed delay period under the criteria of anti-periodic boundaries. It is possible to utilize the piecewise derivative to describe a variety of complex, multi-step, real-world situations that arise from nature. Using fixed point (FP) techniques, like Banach’s FP theorem, Schauder’s FP theorem, and Arzelá Ascoli’s FP theorem, the Hyer–Ulam (HU) stability and the existence theorem conclusions are investigated for the considered problem. Eventually, a supportive example is given to demonstrate the applicability and efficacy of the applied concept. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)
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