Application of Anomalous Diffusion Modeling Based on Fractal and Fractional Derivatives

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: closed (15 May 2024) | Viewed by 3268

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School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Interests: nonlocal/fractional PDEs; high-order algorithms; inverse problems; deep learning
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Guest Editor
School of Sciences, Lanzhou University of Technology, Lanzhou 730000, China
Interests: fractional PDEs; reservoir numerical simulation; deep learning
School of Sciences, Guilin University of Technology, Guilin, China
Interests: efficient numerical methods for nonlocal/fractional PDEs; machine learning and its applications in scientific computing

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Department of Mathematical Sciences, University of South Africa, UNISA, Roodepoort 0003, South Africa
Interests: fractional differential equations; operational matrices; numerical methods; lie symmetry
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Special Issue Information

Dear Colleagues,

Nonlocal diffusion problems have been used to model very different scientific phenomena occurring in various applied fields, for example, in physical, chemistry, biology, fluid dynamics, particle systems, image processing, coagulation models, mathematical finance, etc. In recent years, there has been an explosion of research activity related to numerical methods for nonlocal/fractional differential equations.

The aim of this Special Issue is to collect original and high-quality contributions related to the recent advances in applications of anomalous diffusion modeling as well as efficient numerical methods to simulate the related mathematical, and especially fractional-order, models. Topics that are invited for submission include (but are not limited to):

Analysis and computation using anomalous diffusion modeling;

Applications of nonlocal/fractional differential models;

Machine learning and its applications in scientific computing.

Dr. Minghua Chen
Dr. Jianxiong Cao 
Dr. An Chen
Prof. Dr. H Jafari
Guest Editors

Manuscript Submission Information

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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

  • anomalous diffusion process
  • non-ergodic fractional dynamical systems
  • nonlocal and fractional models
  • fractional biology model
  • non-Gaussian noises
  • fractional stochastic differential equations
  • integral equations
  • finite difference methods
  • finite element methods
  • mathematical models for COVID-19
  • spectral methods
  • fractional lie symmetry
  • local fractional derivative

Published Papers (3 papers)

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Research

14 pages, 12604 KiB  
Article
Adaptive-Coefficient Finite Difference Frequency Domain Method for Solving Time-Fractional Cattaneo Equation with Absorbing Boundary Condition
by Wenhao Xu, Jing Ba, Jianxiong Cao and Cong Luo
Fractal Fract. 2024, 8(3), 146; https://doi.org/10.3390/fractalfract8030146 - 29 Feb 2024
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Abstract
The time-fractional Cattaneo (TFC) equation is a practical tool for simulating anomalous dynamics in physical diffusive processes. The existing numerical solutions to the TFC equation generally deal with the Dirichlet boundary conditions. In this paper, we incorporate the absorbing boundary condition as a [...] Read more.
The time-fractional Cattaneo (TFC) equation is a practical tool for simulating anomalous dynamics in physical diffusive processes. The existing numerical solutions to the TFC equation generally deal with the Dirichlet boundary conditions. In this paper, we incorporate the absorbing boundary condition as a complex-frequency-shifted (CFS) perfectly matched layer (PML) into the TFC equation. Then, we develop an adaptive-coefficient (AC) finite-difference frequency-domain (FDFD) method for solving the TFC with CFS PML. The corresponding analytical solution for homogeneous TFC equation with a point source is proposed for validation. The effectiveness of the developed AC FDFD method is verified by the numerical examples of four typical TFC models, including the different orders of time-fractional derivatives for both the homogeneous model and the layered model. The numerical examples show that the developed AC FDFD method is more accurate than the traditional second-order FDFD method for solving the TFC equation with the CFS PML absorbing boundary condition, while requiring similar computational costs. Full article
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11 pages, 303 KiB  
Article
On the Existence and Ulam Stability of BVP within Kernel Fractional Time
by Hicham Saber, Moheddine Imsatfia, Hamid Boulares, Abdelkader Moumen and Tariq Alraqad
Fractal Fract. 2023, 7(12), 852; https://doi.org/10.3390/fractalfract7120852 - 29 Nov 2023
Viewed by 724
Abstract
This manuscript, we establish novel findings regarding the existence of solutions for second-order fractional differential equations employing Ψ-Caputo fractional derivatives. The application of Banach’s fixed-point theorem (BFPT) ensures the uniqueness of the solutions, while Schauder’s fixed-point theorem (SFPT) is instrumental in determining [...] Read more.
This manuscript, we establish novel findings regarding the existence of solutions for second-order fractional differential equations employing Ψ-Caputo fractional derivatives. The application of Banach’s fixed-point theorem (BFPT) ensures the uniqueness of the solutions, while Schauder’s fixed-point theorem (SFPT) is instrumental in determining the existence of these solutions. Furthermore, we assess the stability of the proposed equation using the Ulam–Hyers stability criterion. To illustrate our results, we provide a concrete example showcasing their practical implications. Full article
41 pages, 619 KiB  
Article
Local Fuzzy Fractional Partial Differential Equations in the Realm of Fractal Calculus with Local Fractional Derivatives
by Mawia Osman, Muhammad Marwan, Syed Omar Shah, Lamia Loudahi, Mahvish Samar, Ebrima Bittaye and Altyeb Mohammed Mustafa
Fractal Fract. 2023, 7(12), 851; https://doi.org/10.3390/fractalfract7120851 - 29 Nov 2023
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Abstract
In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local [...] Read more.
In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration method (LFFSVIM); these are applied when solving LFFPDEs. The working procedure shows how effective solutions for specific LFFPDEs can be obtained using the applied approaches. Moreover, we present a comparison of the local fuzzy fractional Laplace variational iteration method (LFFLIM), the local fuzzy fractional series expansion method (LFFSEM), the local fuzzy fractional variation iteration method (LFFVIM), and the local fuzzy fractional Adomian decomposition method (LFFADM), which are applied to obtain fuzzy fractional diffusion and wave equations on Cantor sets. To demonstrate the effectiveness of the used techniques, some examples are given. The results demonstrate the major advantages of the approaches, which are equally efficient and simple to use in order to solve fuzzy differential equations with local fractional derivatives. Full article
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