Fractal and Fractional

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15 pages, 979 KiB

Open AccessArticle

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 1221

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

(This article belongs to the Special Issue Fractional Diffusion Equations: Numerical Analysis, Modeling and Application)

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21 pages, 10065 KiB

Open AccessArticle

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 - 2 Nov 2023

Viewed by 1034

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

(This article belongs to the Special Issue Fractional Diffusion Equations: Numerical Analysis, Modeling and Application)

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14 pages, 344 KiB

Open AccessArticle

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 - 8 Oct 2023

Viewed by 928

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 + τ^{3 - α} + h^{4})

. 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

(This article belongs to the Special Issue Fractional Diffusion Equations: Numerical Analysis, Modeling and Application)

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11 pages, 329 KiB

Open AccessArticle

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 1241

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

H_{2}^{n} [0, 1]

and the corresponding orthonormal bases are constructed based on scale functions. Then, the space

H_{2, 0}^{2} [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

(This article belongs to the Special Issue Fractional Diffusion Equations: Numerical Analysis, Modeling and Application)

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16 pages, 3052 KiB

Open AccessArticle

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 - 5 Jan 2023

Cited by 7 | Viewed by 1331

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

(This article belongs to the Special Issue Fractional Diffusion Equations: Numerical Analysis, Modeling and Application)

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14 pages, 990 KiB

Open AccessArticle

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 1323

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

(This article belongs to the Special Issue Fractional Diffusion Equations: Numerical Analysis, Modeling and Application)

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18 pages, 1437 KiB

Open AccessArticle

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 1414

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

(This article belongs to the Special Issue Fractional Diffusion Equations: Numerical Analysis, Modeling and Application)

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11 pages, 8153 KiB

Open AccessArticle

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 - 1 Sep 2022

Cited by 21 | Viewed by 1223

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

(This article belongs to the Special Issue Fractional Diffusion Equations: Numerical Analysis, Modeling and Application)

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