Fractional Diffusion Equations: Numerical Analysis, Modeling and Application

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (30 September 2023) | Viewed by 9760

Special Issue Editors


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Guest Editor
School of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Interests: numerical solution of partial differential equations; image-processing technology; nonlinear reaction diffusion equations and their applications; AI and big data processing
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Guest Editor
Department of mathematics and statistics, Changshu Institute of Technology, Suzhou 215500, China
Interests: numerical analysis in the reproducing kernel Hilbert space; numerical analysis of fractional differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Differential equations with fractional-order derivatives have important applications in physics, chemistry, control systems, signal processing, etc. Fractional diffusion models are fundamental mathematical models for the evolution of probability densities. Analytical methods for solving such equations are rarely effective, so it is often necessary to use numerical methods.

This Special Issue will be devoted to collecting recent results on theory, numerical methods and application of fractional diffusion equations and other fractional differential equations. Topics that are invited for submission include (but are not limited to):

  • Theoretical results and numerical methods for fractional diffusion equations;
  • Application of fractional diffusion equations;
  • Numerical methods for fractional oscillating differential equations;
  • Approximation methods for nonsmooth functions;
  • Numerical methods for singular integral equations;
  • Models for fractional differential equations;
  • Theory and numerical methods for fractional-order system identification;
  • Application of fractional-order system identification.

Prof. Dr. Boying Wu
Prof. Dr. Xiuying Li
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

  • fractional diffusion equations
  • fractional oscillating differential equations
  • nonsmooth functions
  • singular integral equations
  • fractional-order system identification
  • modeling
  • application

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Published Papers (8 papers)

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Research

15 pages, 979 KiB  
Article
Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains
by Zhipeng Li, Hongwu Tang, Saiyu Yuan, Huiming Zhang, Lingzhong Kong and HongGuang Sun
Fractal Fract. 2023, 7(11), 823; https://doi.org/10.3390/fractalfract7110823 - 15 Nov 2023
Cited by 1 | Viewed by 1035
Abstract
Recent studies have emphasized the importance of the long-distance diffusion model in characterizing tracer transport occurring within both subsurface and surface environments, particularly in heterogeneous systems. Long-distance diffusion, often referred to as nonlocal diffusion, signifies that tracer particles may experience a considerably long [...] Read more.
Recent studies have emphasized the importance of the long-distance diffusion model in characterizing tracer transport occurring within both subsurface and surface environments, particularly in heterogeneous systems. Long-distance diffusion, often referred to as nonlocal diffusion, signifies that tracer particles may experience a considerably long distance in either the forward or backward direction along preferential channels during the transport. The classical advection–diffusion (ADE) model has been widely used to describe tracer transport; however, they often fall short in capturing the intricacies of nonlocal diffusion processes. The fractional operator has gained recognition among hydrologists due to its potential to capture distinct mechanisms of transport and storage for tracer particles exhibiting nonlocal dynamics. However, the hypersingularity of the fractional Laplacian operator presents considerable difficulties in its numerical approximation in bounded domains. This study focuses on the development and application of the fractional Laplacian-based model to characterize nonlocal tracer transport behavior, specifically considering both forward and backward diffusion processes on bounded domains. The Riesz fractional Laplacian provides a mathematical framework for describing tracer diffusion processes on a bounded domain, and a novel finite difference method (FDM) is introduced as an effective numerical solver for addressing the fractional Laplacian-based model. Applications reveal that the fractional Laplacian-based model can effectively capture the observed nonlocal tracer transport behavior in a heterogeneous system, and nonlocal tracer transport exhibits dynamic characteristics, influenced by the evolving heterogeneity of the media at various temporal scales. Full article
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21 pages, 10065 KiB  
Article
Parallel Algorithm for Solving the Inverse Two-Dimensional Fractional Diffusion Problem of Identifying the Source Term
by Elena N. Akimova, Murat A. Sultanov, Vladimir E. Misilov and Yerkebulan Nurlanuly
Fractal Fract. 2023, 7(11), 801; https://doi.org/10.3390/fractalfract7110801 - 02 Nov 2023
Viewed by 915
Abstract
This paper is devoted to the development of a parallel algorithm for solving the inverse problem of identifying the space-dependent source term in the two-dimensional fractional diffusion equation. For solving the inverse problem, the regularized iterative conjugate gradient method is used. At each [...] Read more.
This paper is devoted to the development of a parallel algorithm for solving the inverse problem of identifying the space-dependent source term in the two-dimensional fractional diffusion equation. For solving the inverse problem, the regularized iterative conjugate gradient method is used. At each iteration of the method, we need to solve the auxilliary direct initial-boundary value problem. By using the finite difference scheme, this problem is reduced to solving a large system of a linear algebraic equation with a block-tridiagonal matrix at each time step. Solving the system takes almost the entire computation time. To solve this system, we construct and implement the direct parallel matrix sweep algorithm. We establish stability and correctness for this algorithm. The parallel implementations are developed for the multicore CPU using the OpenMP technology. The numerical experiments are performed to study the performance of parallel implementations. Full article
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14 pages, 344 KiB  
Article
Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration Problem
by Xinxin Sun, Ailing Zhu, Zhe Yin and Pengfei Ji
Fractal Fract. 2023, 7(10), 739; https://doi.org/10.3390/fractalfract7100739 - 08 Oct 2023
Viewed by 824
Abstract
In this paper, the vibration problem of a beam with a time fractional damping term is studied by the Hermite finite element method, and its fully discrete scheme is obtained. The stability and error estimation of the scheme are analyzed, and it was [...] Read more.
In this paper, the vibration problem of a beam with a time fractional damping term is studied by the Hermite finite element method, and its fully discrete scheme is obtained. The stability and error estimation of the scheme are analyzed, and it was proved that it is unconditionally stable and has a convergence order of O(τ+τ3α+h4). The validity of the scheme is verified by numerical examples, the effects of fractional derivative order and damping coefficient on beam vibration are analyzed and the superiority of the fractional order model has been demonstrated by comparing with the traditional damping model. Full article
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11 pages, 329 KiB  
Article
An ε-Approximate Approach for Solving Variable-Order Fractional Differential Equations
by Yahong Wang, Wenmin Wang, Liangcai Mei, Yingzhen Lin and Hongbo Sun
Fractal Fract. 2023, 7(1), 90; https://doi.org/10.3390/fractalfract7010090 - 13 Jan 2023
Cited by 1 | Viewed by 1132
Abstract
As a mathematical tool, variable-order (VO) fractional calculus (FC) was developed rapidly in the engineering field due to it better describing the anomalous diffusion problems in engineering; thus, the research of the solutions of VO fractional differential equations (FDEs) has become a hot [...] Read more.
As a mathematical tool, variable-order (VO) fractional calculus (FC) was developed rapidly in the engineering field due to it better describing the anomalous diffusion problems in engineering; thus, the research of the solutions of VO fractional differential equations (FDEs) has become a hot topic for the FC community. In this paper, we propose an effective numerical method, named as the ε-approximate approach, based on the least squares theory and the idea of residuals, for the solutions of VO-FDEs and VO fractional integro-differential equations (VO-FIDEs). First, the VO-FDEs and VO-FIDEs are considered to be analyzed in appropriate Sobolev spaces H2n[0,1] and the corresponding orthonormal bases are constructed based on scale functions. Then, the space H2,02[0,1] is chosen which is just suitable for one of the models the authors want to solve to demonstrate the algorithm. Next, the numerical scheme is given, and the stability and convergence are discussed. Finally, four examples with different characteristics are shown, which reflect the accuracy, effectiveness, and wide application of the algorithm. Full article
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16 pages, 3052 KiB  
Article
Fractional Dual-Phase-Lag Model for Nonlinear Viscoelastic Soft Tissues
by Mohamed Abdelsabour Fahmy and Mohammed M. Almehmadi
Fractal Fract. 2023, 7(1), 66; https://doi.org/10.3390/fractalfract7010066 - 05 Jan 2023
Cited by 5 | Viewed by 1157
Abstract
The primary goal of this paper is to create a new fractional boundary element method (BEM) model for bio-thermomechanical problems in functionally graded anisotropic (FGA) nonlinear viscoelastic soft tissues. The governing equations of bio-thermomechanical problems are briefly presented, including the fractional dual-phase-lag (DPL) [...] Read more.
The primary goal of this paper is to create a new fractional boundary element method (BEM) model for bio-thermomechanical problems in functionally graded anisotropic (FGA) nonlinear viscoelastic soft tissues. The governing equations of bio-thermomechanical problems are briefly presented, including the fractional dual-phase-lag (DPL) bioheat model and Biot’s model. The more complex shapes of nonlinear viscoelastic soft tissues can be handled by the boundary element method, which also avoids the need for the interior domain to be discretized. The fractional dual-phase-lag bioheat equation was solved using the general boundary element method (GBEM) based on the local radial basis function collocation method (LRBFCM). The poroelastic fields are then calculated using the convolution quadrature boundary element method (CQBEM) The numerical findings show that our proposed numerical model is valid, efficient, and accurate. Full article
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14 pages, 990 KiB  
Article
Finite Element Approximations to Caputo–Hadamard Time-Fractional Diffusion Equation with Application in Parameter Identification
by Shijing Cheng, Ning Du, Hong Wang and Zhiwei Yang
Fractal Fract. 2022, 6(9), 525; https://doi.org/10.3390/fractalfract6090525 - 17 Sep 2022
Viewed by 1189
Abstract
A finite element scheme for solving a two-timescale Hadamard time-fractional equation is discussed. We prove the error estimate without assuming the smoothness of the solution. In order to invert the fractional order, a finite-element Levenberg–Marquardt method is designed. Finally, we give corresponding numerical [...] Read more.
A finite element scheme for solving a two-timescale Hadamard time-fractional equation is discussed. We prove the error estimate without assuming the smoothness of the solution. In order to invert the fractional order, a finite-element Levenberg–Marquardt method is designed. Finally, we give corresponding numerical experiments to support the correctness of our analysis. Full article
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18 pages, 1437 KiB  
Article
A Mixed Finite Volume Element Method for Time-Fractional Damping Beam Vibration Problem
by Tongxin Wang, Ziwen Jiang, Ailing Zhu and Zhe Yin
Fractal Fract. 2022, 6(9), 523; https://doi.org/10.3390/fractalfract6090523 - 16 Sep 2022
Cited by 4 | Viewed by 1275
Abstract
In this paper, the transverse vibration of a fractional viscoelastic beam is studied based on the fractional calculus, and the corresponding scheme of a viscoelastic beam is established by using the mixed finite volume element method. The stability and convergence of the algorithm [...] Read more.
In this paper, the transverse vibration of a fractional viscoelastic beam is studied based on the fractional calculus, and the corresponding scheme of a viscoelastic beam is established by using the mixed finite volume element method. The stability and convergence of the algorithm are analyzed. Numerical examples demonstrate the effectiveness of the algorithm. Finally, the values of different parameter sets are tested, and the test results show that both the damping coefficient and the fractional derivative have significant effects on the model. The results of this paper can be used for the damping modeling of viscoelastic structures. Full article
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11 pages, 8153 KiB  
Article
Solving Two-Sided Fractional Super-Diffusive Partial Differential Equations with Variable Coefficients in a Class of New Reproducing Kernel Spaces
by Zhiyuan Li, Qintong Chen, Yulan Wang and Xiaoyu Li
Fractal Fract. 2022, 6(9), 492; https://doi.org/10.3390/fractalfract6090492 - 01 Sep 2022
Cited by 18 | Viewed by 1100
Abstract
Fractional-order calculus has become a useful mathematical framework to describe the complex super-diffusive process; however, numerical solutions of the two-sided space-fractional super-diffusive model with variable coefficients are difficult to obtain, and almost no method can obtain an analytical solution. In this paper, a [...] Read more.
Fractional-order calculus has become a useful mathematical framework to describe the complex super-diffusive process; however, numerical solutions of the two-sided space-fractional super-diffusive model with variable coefficients are difficult to obtain, and almost no method can obtain an analytical solution. In this paper, a class of new fractional dimensional reproducing kernel spaces (RKS) based on Caputo fractional derivatives is given, and we give analytical and numerical solutions of the two-sided space-fractional super-diffusive model based on the class of new RKS. The analytical solution is represented in the form of series in the reproducing kernel space. Numerical experiments indicate that the piecewise reproducing kernel method is more accurate than the traditional reproducing kernel method (RKM), and these new fractional reproducing kernel spaces are efficient for the two-sided space-fractional super-diffusive model. Full article
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