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Fractal Fract
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Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition
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Numerical Solution and Applications of Fractional Differential Equations, 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: 25 December 2025 | Viewed by 3308

Special Issue Editors

Dr. Libo Feng

E-Mail Website
Guest Editor
School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, QLD 4001, Australia
Interests: numerical methods and analysis of fractional PDE; application of fractional mathematical models
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Yang Liu

E-Mail Website
Guest Editor
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
Interests: finite element method; finite difference method; LDG methods; numerical methods for fractional PDEs
Special Issues, Collections and Topics in MDPI journals
Dr. Lin Liu

E-Mail Website
Guest Editor
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
Interests: viscoelastic fluid boundary layer flow; fractional anomalous diffusion; biological heat conduction
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Over the last few decades, the application of fractional calculus to real-world problems has grown rapidly, in which the use of dynamical systems described by fractional differential equations (FDEs) had been one of the ways to understand complex materials and processes. Due to the power to model the non-locality, memory, spatial heterogeneity, and anomalous diffusion inherent in many real-world problems, the application of FDEs has been attracting much attention in many fields of science and is still under development. However, generally, the fractional mathematical models from science and engineering are so complex that analytical solutions are not available. Therefore, the numerical solution has been an effective tool to deal with fractional mathematical models.

This Special Issue aims to promote communication between researchers and practitioners on the application of fractional calculus, present the latest development of fractional differential equations, report state-of-the-art and in-progress numerical methods, and discuss future trends and challenges. We cordially invite you to contribute by submitting original research articles or comprehensive review papers. This Special Issue will cover, but is not limited to, the following topics:

  • Mathematical modeling of fractional dynamic systems;
  • Analytical or semi-analytical solution of fractional differential equations;
  • Numerical methods to solve fractional differential equations, e.g., the finite difference method, the finite element method, the finite volume method, the spectral method, etc.;
  • Fast algorithm for the time- or space-fractional derivative;
  • Mathematical analysis for fractional problems and numerical analysis for the numerical scheme;
  • Applications of fractional calculus in physics, biology, chemistry, finance, signal and image processing, hydrology, non-Newtonian fluids, etc.

You are also welcome to read and download all published articles in the first edition: https://www.mdpi.com/journal/fractalfract/special_issues/NSAFDE, and the second: https://www.mdpi.com/journal/fractalfract/special_issues/NSAFDE_II.

Dr. Libo Feng
Prof. Dr. Yang Liu
Dr. Lin Liu
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 250 words) can be sent to the Editorial Office for assessment.

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

  • numerical methods
  • mathematical modelling
  • fractional calculus
  • fractional differential equations
  • numerical analysis
  • fast algorithm

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

Related Special Issues

Published Papers (4 papers)

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Research

23 pages, 4068 KB  
Article
Numerical Treatment of the Time-Fractional Kuramoto–Sivashinsky Equation Using a Combined Chebyshev-Collocation Approach
by Waleed Mohamed Abd-Elhameed, Mohamed A. Abdelkawy, Naher Mohammed A. Alsafri and Ahmed Gamal Atta
Fractal Fract. 2025, 9(11), 727; https://doi.org/10.3390/fractalfract9110727 - 10 Nov 2025
Viewed by 388
Abstract
In this paper, we present a collocation algorithm for numerically treating the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Certain orthogonal polynomials, which are expressed as combinations of Chebyshev polynomials, and their shifted polynomials are introduced. Some new theoretical formulas regarding these polynomials have been developed, [...] Read more.
In this paper, we present a collocation algorithm for numerically treating the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Certain orthogonal polynomials, which are expressed as combinations of Chebyshev polynomials, and their shifted polynomials are introduced. Some new theoretical formulas regarding these polynomials have been developed, including their operational matrices of both integer and fractional derivatives. The derived formulas will be the foundation for designing the proposed numerical algorithm, which relies on converting the governing problem with its underlying conditions into a nonlinear algebraic system, which can be solved using Newton’s iteration technique. A rigorous error analysis for the proposed combined Chebyshev expansion is presented. Some numerical examples are given to ensure the applicability and efficiency of the presented algorithm. These results demonstrate that the proposed algorithm attains superior accuracy with fewer expansion terms. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)
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32 pages, 1899 KB  
Article
A Physics-Informed Neural Network Based on the Separation of Variables for Solving the Distributed-Order Time-Fractional Advection–Diffusion Equation
by Wenkai Liu and Yang Liu
Fractal Fract. 2025, 9(11), 712; https://doi.org/10.3390/fractalfract9110712 - 4 Nov 2025
Viewed by 930
Abstract
In this work, we propose a new physics-informed neural network framework based on the method of separation of variables (SVPINN) to solve the distributed-order time-fractional advection–diffusion equation. We develop a new method for calculating the distributed-order derivative, which enables the fractional integral to [...] Read more.
In this work, we propose a new physics-informed neural network framework based on the method of separation of variables (SVPINN) to solve the distributed-order time-fractional advection–diffusion equation. We develop a new method for calculating the distributed-order derivative, which enables the fractional integral to be modeled by a network and directly solved by combining automatic differentiation technology. In this way, the approximation of the distributed-order derivative is integrated into the parameter training system of the network, and the data-driven adaptive learning mechanism is used to replace the numerical discretization scheme. In the SVPINN framework, we decompose the kernel function of the Caputo integral into three independent functions using the method of separation of variables, and apply a neural network as a surrogate model for the modified integral and the function related to the time variable. The new physical constraint generated by the modified integral serves as an extra supervised learning task for the network. We systematically evaluated the feasibility of the SVPINN on several numerical experiments and demonstrated its performance. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)
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28 pages, 4465 KB  
Article
Neural Networks-Based Analytical Solver for Exact Solutions of Fractional Partial Differential Equations
by Shanhao Yuan, Yanqin Liu, Limei Yan, Runfa Zhang and Shunjun Wu
Fractal Fract. 2025, 9(8), 541; https://doi.org/10.3390/fractalfract9080541 - 16 Aug 2025
Cited by 1 | Viewed by 1031
Abstract
This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in [...] Read more.
This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in both computational speed and solution precision. The efficacy of the proposed method is validated through four numerical examples, with results visualized using three-dimensional surface plots, contour mappings, and density distributions. Numerical experiments demonstrate that the proposed framework successfully derives exact solutions for fPDEs without relying on data samples. This research provides a novel methodological framework for solving fPDEs, with broad applicability across scientific and engineering fields. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)
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20 pages, 873 KB  
Article
A Mixed Finite Volume Element Method for Nonlinear Time Fractional Fourth-Order Reaction–Diffusion Models
by Jie Zhao, Min Cao and Zhichao Fang
Fractal Fract. 2025, 9(8), 481; https://doi.org/10.3390/fractalfract9080481 - 23 Jul 2025
Cited by 1 | Viewed by 588
Abstract
In this paper, a linearized mixed finite volume element (MFVE) scheme is proposed to solve the nonlinear time fractional fourth-order reaction–diffusion models with the Riemann–Liouville time fractional derivative. By introducing an auxiliary variable σ=Δu, the original fourth-order model is [...] Read more.
In this paper, a linearized mixed finite volume element (MFVE) scheme is proposed to solve the nonlinear time fractional fourth-order reaction–diffusion models with the Riemann–Liouville time fractional derivative. By introducing an auxiliary variable σ=Δu, the original fourth-order model is reformulated into a lower-order coupled system. The first-order time derivative and the time fractional derivative are discretized by using the BDF2 formula and the weighted and shifted Grünwald difference (WSGD) formula, respectively. Then, a fully discrete MFVE scheme is constructed by using the primal and dual grids. The existence and uniqueness of a solution for the MFVE scheme are proven based on the matrix theories. The scheme’s unconditional stability is rigorously derived by using the Gronwall inequality in detail. Moreover, the optimal error estimates for u in the discrete L∞(L2(Ω)) and L2(H1(Ω)) norms and for σ in the discrete L2(L2(Ω)) norm are obtained. Finally, three numerical examples are given to confirm its feasibility and effectiveness. Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)
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