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Fractal Fract
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Recent Advances in Fractional Differential Equations and Their Applications, 3rd...
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Recent Advances in Fractional Differential Equations and Their Applications, 3rd 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 2025 | Viewed by 167

Special Issue Editors

Prof. Dr. Xiaoli Chen

E-Mail Website
Guest Editor
Department of Information and Computing Sciences, China University of Geosciences, Wuhan, China
Interests: fractional partial differential equation; machine learning; stochastic dynamical systems
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 Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional differential equations describe the dynamic systems of complex and nonlocal 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 represents a pressing issue.

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

  • 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 estimates and homogenization for nonlocal differential equations;
  • Application of stochastic dynamics and fractional models;
  • Machine learning methods for FDEs;
  • Inverse problems in nonlocal PDE / SDE;
  • Effective dynamics and reduced-order models.

Please feel free to read and download all the articles published in our first volume:

https://www.mdpi.com/journal/fractalfract/special_issues/222BH46HIW

Our second volume is also available:

https://www.mdpi.com/journal/fractalfract/special_issues/fractional_differential_equations2

We look forward to receiving your submissions.

Prof. 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 equations
  • fractional differential equations
  • stochastic differential equations
  • numerical method
  • machine learning methods

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

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Published Papers (1 paper)

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Research

18 pages, 4917 KiB  
Article
Rapid Estimation of Soil Copper Content Using a Novel Fractional Derivative Three-Band Index and Spaceborne Hyperspectral Data
by Shichao Cui, Guo Jiang and Jiawei Lu
Fractal Fract. 2025, 9(8), 523; https://doi.org/10.3390/fractalfract9080523 - 12 Aug 2025
Abstract
Rapid and large-scale monitoring of soil copper levels enables the quick identification of areas where copper concentrations significantly exceed safe thresholds. It allows for selecting regions that require treatment and protection and is essential for safeguarding environmental and human health. Widely adopted monitoring [...] Read more.
Rapid and large-scale monitoring of soil copper levels enables the quick identification of areas where copper concentrations significantly exceed safe thresholds. It allows for selecting regions that require treatment and protection and is essential for safeguarding environmental and human health. Widely adopted monitoring models that utilize ground- and uncrewed-aerial-vehicle-based spectral data are superior to labor-intensive and time-consuming traditional methods that rely on point sampling, chemical analysis, and spatial interpolation. However, these methods are unsuitable for large-scale observations. Therefore, this study investigates the potential of utilizing spaceborne GF-5 hyperspectral data for monitoring soil copper content. Ninety-five soil samples were collected from the Kalatage mining area in Xinjiang, China. Three-band indices were constructed using fractional derivative spectra, and estimation models were developed using spectral indices highly correlated with the copper content. The results show that the proposed three-band spectral index accurately identifies subtle spectral characteristics associated with the copper content. Although the model is relatively simple, selecting the correct fractional order is critical in constructing spectral indices. The three-band spectral index based on fractional derivatives with orders of less than 0.6 provides higher accuracy than higher-order fractional derivatives. The index with spectral wavelengths of 426.796 nm, 512.275 nm, and 974.245 nm with 0.35-order derivatives exhibits the optimal performance (R2 = 0.51, RPD = 1.46). Additionally, we proposed a novel approach that identifies the three-band indices exhibiting a strong correlation with the copper content. Subsequently, the selected indices were used as independent variables to develop new spectral indices for model development. This approach provides higher performance than models that use spectral indices derived from individual band values. The model utilizing the proposed spectral index achieved the best performance (R2 = 0.56, RPD = 1.52). These results indicate that utilizing GF-5 hyperspectral data for large-scale monitoring of soil copper content is feasible and practical. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 3rd Edition)
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