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 May 2024 | Viewed by 4364

Special Issue Editors


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Guest Editor
Department of Mathematics, National University of Singapore, Singapore, Singapore
Interests: fractional partial differential equation; machine learning; stochastic dynamical systems
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Guest Editor

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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

The first volume of this Special Issue was a great success. The published articles can be read at: https://www.mdpi.com/journal/fractalfract/special_issues/222BH46HIW

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

Published Papers (4 papers)

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Research

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 1020
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
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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
Viewed by 979
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
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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 691
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
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 4 | Viewed by 1024
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
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