Numerical Solution and Applications of Fractional Differential Equations, 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: 30 June 2024 | Viewed by 1915

Special Issue Editors

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
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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
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,

In the last few decades, the application of fractional calculus to real-world problems has grown rapidly, with the use of dynamical systems described by fractional differential equations (FDEs) as one of the ways to understand complex materials and processes. Due to the ability 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, 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 the following topics, but these are not exhaustive:

  1. Mathematical modeling of fractional dynamic systems;
  2. Analytical or semi-analytical solution of fractional differential equations;
  3. 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.;
  4. Fast algorithm for the time or space fractional derivative;
  5. Mathematical analysis for fractional problems and numerical analysis for the numerical scheme;
  6. Applications of fractional calculus in physics, biology, chemistry, finance, signal and image processing, hydrology, non-Newtonian fluids, etc.

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

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

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 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

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

Published Papers (3 papers)

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Research

19 pages, 1361 KiB  
Article
Dynamical Analysis of Two-Dimensional Fractional-Order-in-Time Biological Population Model Using Chebyshev Spectral Method
by Ishtiaq Ali
Fractal Fract. 2024, 8(6), 325; https://doi.org/10.3390/fractalfract8060325 (registering DOI) - 29 May 2024
Viewed by 15
Abstract
In this study, we investigate the application of fractional calculus to the mathematical modeling of biological systems, focusing on fractional-order-in-time partial differential equations (FTPDEs). Fractional derivatives, especially those defined in the Caputo sense, provide a useful tool for modeling memory and hereditary characteristics, [...] Read more.
In this study, we investigate the application of fractional calculus to the mathematical modeling of biological systems, focusing on fractional-order-in-time partial differential equations (FTPDEs). Fractional derivatives, especially those defined in the Caputo sense, provide a useful tool for modeling memory and hereditary characteristics, which are problems that are frequently faced with integer-order models. We use the Chebyshev spectral approach for spatial derivatives, which is known for its faster convergence rate, in conjunction with the L1 scheme for time-fractional derivatives because of its high accuracy and robustness in handling nonlocal effects. A detailed theoretical analysis, followed by a number of numerical experiments, is performed to confirmed the theoretical justification. Our simulation results show that our numerical technique significantly improves the convergence rates, effectively tackles computing difficulties, and provides a realistic simulation of biological population dynamics. Full article
22 pages, 8443 KiB  
Article
Fractional Second-Grade Fluid Flow over a Semi-Infinite Plate by Constructing the Absorbing Boundary Condition
by Jingyu Yang, Lin Liu, Siyu Chen, Libo Feng and Chiyu Xie
Fractal Fract. 2024, 8(6), 309; https://doi.org/10.3390/fractalfract8060309 - 23 May 2024
Viewed by 358
Abstract
The modified second-grade fluid flow across a plate of semi-infinite extent, which is initiated by the plate’s movement, is considered herein. The relaxation parameters and fractional parameters are introduced to express the generalized constitutive relation. A convolution-based absorbing boundary condition (ABC) is developed [...] Read more.
The modified second-grade fluid flow across a plate of semi-infinite extent, which is initiated by the plate’s movement, is considered herein. The relaxation parameters and fractional parameters are introduced to express the generalized constitutive relation. A convolution-based absorbing boundary condition (ABC) is developed based on the artificial boundary method (ABM), addressing issues related to the semi-infinite boundary. We adopt the finite difference method (FDM) for deriving the numerical solution by employing the L1 scheme to approximate the fractional derivative. To confirm the precision of this method, a source term is added to establish an exact solution for verification purposes. A comparative evaluation of the ABC versus the direct truncated boundary condition (DTBC) is conducted, with their effectiveness and soundness being visually scrutinized and assessed. This study investigates the impact of the motion of plates at different fluid flow velocities, focusing on the effects of dynamic elements influencing flow mechanisms and velocity. This research’s primary conclusion is that a higher fractional parameter correlates with the fluid flow. As relaxation parameters decrease, the delay effect intensifies and the fluid velocity decreases. Full article
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20 pages, 372 KiB  
Article
A Mixed Finite Element Method for the Multi-Term Time-Fractional Reaction–Diffusion Equations
by Jie Zhao, Shubin Dong and Zhichao Fang
Fractal Fract. 2024, 8(1), 51; https://doi.org/10.3390/fractalfract8010051 - 12 Jan 2024
Viewed by 967
Abstract
In this work, a fully discrete mixed finite element (MFE) scheme is designed to solve the multi-term time-fractional reaction–diffusion equations with variable coefficients by using the well-known L1 formula and the Raviart–Thomas MFE space. The existence and uniqueness of the discrete solution [...] Read more.
In this work, a fully discrete mixed finite element (MFE) scheme is designed to solve the multi-term time-fractional reaction–diffusion equations with variable coefficients by using the well-known L1 formula and the Raviart–Thomas MFE space. The existence and uniqueness of the discrete solution is proved by using the matrix theory, and the unconditional stability is also discussed in detail. By introducing the mixed elliptic projection, the error estimates for the unknown variable u in the discrete L(L2(Ω)) norm and for the auxiliary variable λ in the discrete L((L2(Ω))2) and L(H(div,Ω)) norms are obtained. Finally, three numerical examples are given to demonstrate the theoretical results. Full article
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