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Fractal Fract
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Recent Advances in the Spatial and Temporal Discretizations of Fractional...
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Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs

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 July 2024 | Viewed by 6822

Special Issue Editors

Dr. Xian-Ming Gu

E-Mail Website
Guest Editor
1. School of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China
2. Mathematical Institute, Utrecht University, 3584 Utrecht, The Netherlands
Interests: numerical linear algebra; numerical (fractional) PDEs; parallel-in-time methods; Krylov subspace solvers
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Hongbin Chen

E-Mail Website
Guest Editor
Institute of Mathematics and Physics, College of Science, Central South University of Forestry and Technology, Changsha 410004, China
Interests: finite difference, finite volume and finite element methods for time fractional differential equations; finite element and finite difference methods for integral fractional Laplace
Prof. Dr. Shulin Wu

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
Interests: PDEs/ODEs; neural networks; algorithms; optimization; numerical analysis; applied and computational mathematics
Prof. Dr. Xiangcheng Zheng

E-Mail Website
Guest Editor
School of Mathematics, Shandong University, Jinan 250100, China
Interests: fractional calculus; fractional differential equation; variable-order; numerical method; mathematical analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional PDEs (FPDEs) generalize the classic (integer-order) calculus and PDEs to any differential form of fractional orders. FPDEs are emerging as a powerful tool for modeling challenging multiscale phenomena, including overlapping microscopic and macroscopic scales, anomalous transport and long-range time memory or spatial interactions. However, the exact solutions of FPDEs cannot be explicitly expressed, thus numerical methods based on various spatial and temporal discretizations have become the mainstream tools for such FPDEs and have had a booming development over the past several decades. These spatial and temporal discretizations that maintain the important characteristics or structures of FPDEs, such as weak singularity, optimal long-time decay rate, long-term numerical stability and the convergence of numerical schemes for such FPDEs, are still limited. Therefore, developing efficient spatial and temporal discretizations for the numerical solutions of FPDEs is still quite challenging in the field of numerical analysis.

This Special Issue will provide a platform for the recent and original research results on efficient numerical methods for solving FPDEs. We invite authors to contribute original research articles for the Special Issue “Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs”. The following potential topics include, but are not limited to:

  • Finite difference, finite element, finite volume, spectral methods;
  • Nonuniform and adaptive discretizations;
  • Adaptive space–time methods;
  • Numerical treatments of integro-differential equations;
  • Parallel-in-time methods;
  • Fast matrix computations arising from numerical methods of FPDEs;
  • Nonlocal modeling and computation;
  • Convolution quadrature;
  • Modeling and simulations involving (fractional) PDEs.

Dr. Xian-Ming Gu
Prof. Dr. Hongbin Chen
Prof. Dr. Shulin Wu
Prof. Dr. Xiangcheng Zheng
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

  • fractional PDEs
  • finite difference, finite element, finite volume, spectral methods
  • nonuniform and adaptive discretizations
  • adaptive space-time methods
  • parallel-in-time methods
  • numerical methods
  • numerical treatments of integro-differential equations
  • nonlocal modeling and computation
  • convolution quadrature
  • modeling and simulations

Published Papers (6 papers)

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Research

14 pages, 301 KiB  
Article
A Temporal Second-Order Difference Scheme for Variable-Order-Time Fractional-Sub-Diffusion Equations of the Fourth Order
by Xin Zhang, Yu Bo and Yuanfeng Jin
Fractal Fract. 2024, 8(2), 112; https://doi.org/10.3390/fractalfract8020112 - 13 Feb 2024
Viewed by 958
Abstract
In this article, we develop a compact finite difference scheme for a variable-order-time fractional-sub-diffusion equation of a fourth-order derivative term via order reduction. The proposed scheme exhibits fourth-order convergence in space and second-order convergence in time. Additionally, we provide a detailed proof for [...] Read more.
In this article, we develop a compact finite difference scheme for a variable-order-time fractional-sub-diffusion equation of a fourth-order derivative term via order reduction. The proposed scheme exhibits fourth-order convergence in space and second-order convergence in time. Additionally, we provide a detailed proof for the existence and uniqueness, as well as the stability of scheme, along with a priori error estimates. Finally, we validate our theoretical results through various numerical computations. Full article
(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)
20 pages, 438 KiB  
Article
A Bound-Preserving Numerical Scheme for Space–Time Fractional Advection Equations
by Jing Gao and Huaiguang Chen
Fractal Fract. 2024, 8(2), 89; https://doi.org/10.3390/fractalfract8020089 - 30 Jan 2024
Viewed by 1020
Abstract
We develop and analyze an explicit finite difference scheme that satisfies a bound-preserving principle for space–time fractional advection equations with the orders of 0<α and β≤1. The stability (and convergence) of the method is discussed. Due to the [...] Read more.
We develop and analyze an explicit finite difference scheme that satisfies a bound-preserving principle for space–time fractional advection equations with the orders of 0<α and β≤1. The stability (and convergence) of the method is discussed. Due to the nonlocal property of the fractional operators, the numerical method generates dense coefficient matrices with complex structures. In order to increase the effectiveness of the method, we use Toeplitz-like structures in the full coefficient matrix in a sparse form to reduce the costs of computation, and we also apply a fast evaluation method for the time–fractional derivative. Therefore, an efficient solver is constructed. Numerical experiments are provided for the utility of the method. Full article
(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)
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17 pages, 3749 KiB  
Article
Heavy Tail and Long-Range Dependence for Skewed Time Series Prediction Based on a Fractional Weibull Process
by Wanqing Song, Dongdong Chen and Enrico Zio
Fractal Fract. 2024, 8(1), 7; https://doi.org/10.3390/fractalfract8010007 - 20 Dec 2023
Viewed by 1043
Abstract
In this paper, a fractional Weibull process is utilized in a predictive stochastic differential equation model to allow for skewness and heavy-tailed characteristics. To this aim, a fractional Weibull process with non-Gaussian characteristics and a long memory effect is proposed to drive the [...] Read more.
In this paper, a fractional Weibull process is utilized in a predictive stochastic differential equation model to allow for skewness and heavy-tailed characteristics. To this aim, a fractional Weibull process with non-Gaussian characteristics and a long memory effect is proposed to drive the predictive stochastic differential equation. The difference iterative forecasting model is proposed as its stochastic difference scheme. The consistency, stability, and convergence of the model are analyzed. In the proposed model, variational mode decomposition is utilized as the data preprocessing approach to separate the stationary and non-stationary components. Actual wind speed data and stock price data are employed in two separate case studies. Full article
(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)
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15 pages, 399 KiB  
Article
An Analysis and Global Identification of Smoothless Variable Order of a Fractional Stochastic Differential Equation
by Qiao Li, Xiangcheng Zheng, Hong Wang, Zhiwei Yang and Xu Guo
Fractal Fract. 2023, 7(12), 850; https://doi.org/10.3390/fractalfract7120850 - 29 Nov 2023
Viewed by 867
Abstract
We establish both the uniqueness and the existence of the solutions to a hidden-memory variable-order fractional stochastic partial differential equation, which models, e.g., the stochastic motion of a Brownian particle within a viscous liquid medium varied with fractal dimensions. We also investigate the [...] Read more.
We establish both the uniqueness and the existence of the solutions to a hidden-memory variable-order fractional stochastic partial differential equation, which models, e.g., the stochastic motion of a Brownian particle within a viscous liquid medium varied with fractal dimensions. We also investigate the inverse problem concerning the observations of the solutions, which eliminates the analytic assumptions on the variable orders in the literature of this topic and theoretically guarantees the reliability of the determination and experimental inference. Full article
(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)
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13 pages, 383 KiB  
Article
A Predictor–Corrector Compact Difference Scheme for a Nonlinear Fractional Differential Equation
by Xiaoxuan Jiang, Jiawei Wang, Wan Wang and Haixiang Zhang
Fractal Fract. 2023, 7(7), 521; https://doi.org/10.3390/fractalfract7070521 - 30 Jun 2023
Cited by 36 | Viewed by 1225
Abstract
In this work, a predictor–corrector compact difference scheme for a nonlinear fractional differential equation is presented. The MacCormack method is provided to deal with nonlinear terms, the Riemann–Liouville (R-L) fractional integral term is treated by means of the second-order convolution quadrature formula, and [...] Read more.
In this work, a predictor–corrector compact difference scheme for a nonlinear fractional differential equation is presented. The MacCormack method is provided to deal with nonlinear terms, the Riemann–Liouville (R-L) fractional integral term is treated by means of the second-order convolution quadrature formula, and the Caputo derivative term is discretized by the L1 discrete formula. Through the first and second derivatives of the matrix under the compact difference, we improve the precision of this scheme. Then, the existence and uniqueness are proved, and the numerical experiments are presented. Full article
(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)
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12 pages, 690 KiB  
Article
Interpolated Coefficient Mixed Finite Elements for Semilinear Time Fractional Diffusion Equations
by Xiaowu Li and Yuelong Tang
Fractal Fract. 2023, 7(6), 482; https://doi.org/10.3390/fractalfract7060482 - 16 Jun 2023
Cited by 4 | Viewed by 912
Abstract
In this paper, we consider a fully discrete interpolated coefficient mixed finite element method for semilinear time fractional reaction–diffusion equations. The classic L1 scheme based on graded meshes and new mixed finite element based on triangulation is used for the temporal and [...] Read more.
In this paper, we consider a fully discrete interpolated coefficient mixed finite element method for semilinear time fractional reaction–diffusion equations. The classic L1 scheme based on graded meshes and new mixed finite element based on triangulation is used for the temporal and spatial discretization, respectively. The interpolation coefficient technique is used to deal with the semilinear term, and the discrete nonlinear system is solved by a Newton-like iterative method. Stability and convergence results for both the original variable and its flux are derived. Numerical experiments confirm our theoretical analysis. Full article
(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)
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