Fractal and Fractional

Research

21 pages, 655 KiB

Open AccessArticle

Approximate Solution of a Kind of Time-Fractional Evolution Equations Based on Fast L1 Formula and Barycentric Lagrange Interpolation

by Ting Liu, Hongyan Liu and Yanying Ma

Fractal Fract. 2024, 8(11), 675; https://doi.org/10.3390/fractalfract8110675 - 20 Nov 2024

Viewed by 347

Abstract

In this paper, an effective numerical approach that combines the fast L1 formula and barycentric Lagrange interpolation is proposed for solving a kind of time-fractional evolution equations. This type of equation contains a nonlocal term involving the time variable, resulting in extremely high [...] Read more.

In this paper, an effective numerical approach that combines the fast L1 formula and barycentric Lagrange interpolation is proposed for solving a kind of time-fractional evolution equations. This type of equation contains a nonlocal term involving the time variable, resulting in extremely high computational complexity of numerical discrete formats in general. To reduce the computational burden, the fast L1 technique based on the L1 formula and sum-of-exponentials approximation is employed to evaluate the Caputo time-fractional derivative. Meanwhile, a fast and unconditionally stable time semi-discrete format is obtained. Subsequently, we utilize the barycentric Lagrange interpolation and its differential matrices to achieve spatial discretizations so as to deduce fully discrete formats. Then error estimates of related fully discrete formats are explored. Eventually, some numerical experiments are simulated to testify to the effective and fast behavior of the presented 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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16 pages, 1637 KiB

Open AccessArticle

Neural Fractional Differential Equations: Optimising the Order of the Fractional Derivative

by Cecília Coelho, M. Fernanda P. Costa and Luís L. Ferrás

Fractal Fract. 2024, 8(9), 529; https://doi.org/10.3390/fractalfract8090529 - 10 Sep 2024

Viewed by 870

Abstract

Neural Fractional Differential Equations (Neural FDEs) represent a neural network architecture specifically designed to fit the solution of a fractional differential equation to given data. This architecture combines an analytical component, represented by a fractional derivative, with a neural network component, forming an [...] Read more.

Neural Fractional Differential Equations (Neural FDEs) represent a neural network architecture specifically designed to fit the solution of a fractional differential equation to given data. This architecture combines an analytical component, represented by a fractional derivative, with a neural network component, forming an initial value problem. During the learning process, both the order of the derivative and the parameters of the neural network must be optimised. In this work, we investigate the non-uniqueness of the optimal order of the derivative and its interaction with the neural network component. Based on our findings, we perform a numerical analysis to examine how different initialisations and values of the order of the derivative (in the optimisation process) impact its final optimal value. Results show that the neural network on the right-hand side of the Neural FDE struggles to adjust its parameters to fit the FDE to the data dynamics for any given order of the fractional derivative. Consequently, Neural FDEs do not require a unique

α

value; instead, they can use a wide range of

α

values to fit data. This flexibility is beneficial when fitting to given data is required and the underlying physics is not known. 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, 719 KiB

Open AccessArticle

Difference Approximation for 2D Time-Fractional Integro-Differential Equation with Given Initial and Boundary Conditions

by Xindong Zhang, Ziyang Luo, Quan Tang, Leilei Wei and Juan Liu

Fractal Fract. 2024, 8(8), 495; https://doi.org/10.3390/fractalfract8080495 - 22 Aug 2024

Viewed by 668

Abstract

In this investigation, a new algorithm based on the compact difference method is proposed. The purpose of this investigation is to solve the 2D time-fractional integro-differential equation. The Riemann–Liouville derivative was utilized to define the time-fractional derivative. Meanwhile, the weighted and shifted Grünwald [...] Read more.

In this investigation, a new algorithm based on the compact difference method is proposed. The purpose of this investigation is to solve the 2D time-fractional integro-differential equation. The Riemann–Liouville derivative was utilized to define the time-fractional derivative. Meanwhile, the weighted and shifted Grünwald difference operator and product trapezoidal formula were utilized to construct a high-order numerical scheme. Also, we analyzed the stability and convergence. The convergence order was

O (τ^{2} + h_{x}^{4} + h_{y}^{4})

, where

τ

is the time step size,

h_{x}

and

h_{y}

are the spatial step sizes. Furthermore, several examples were provided to verify the correctness of our theoretical reasoning. 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, 1786 KiB

Open AccessArticle

Numerical Analysis and Computation of the Finite Volume Element Method for the Nonlinear Coupled Time-Fractional Schrödinger Equations

by Xinyue Zhao, Yining Yang, Hong Li, Zhichao Fang and Yang Liu

Fractal Fract. 2024, 8(8), 480; https://doi.org/10.3390/fractalfract8080480 - 17 Aug 2024

Viewed by 586

Abstract

In this article, our aim is to consider an efficient finite volume element method combined with the

L 2 - 1_{σ}

formula for solving the coupled Schrödinger equations with nonlinear terms and time-fractional derivative terms. We design the fully discrete scheme, where [...] Read more.

In this article, our aim is to consider an efficient finite volume element method combined with the

L 2 - 1_{σ}

formula for solving the coupled Schrödinger equations with nonlinear terms and time-fractional derivative terms. We design the fully discrete scheme, where the space direction is approximated using the finite volume element method and the time direction is discretized making use of the

L 2 - 1_{σ}

formula. We then prove the stability for the fully discrete scheme, and derive the optimal convergence result, from which one can see that our scheme has second-order accuracy in both the temporal and spatial directions. We carry out numerical experiments with different examples to verify the optimal convergence result. Full article

(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)

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21 pages, 405 KiB

Open AccessArticle

A Fast Second-Order ADI Finite Difference Scheme for the Two-Dimensional Time-Fractional Cattaneo Equation with Spatially Variable Coefficients

by Lijuan Nong, Qian Yi and An Chen

Fractal Fract. 2024, 8(8), 453; https://doi.org/10.3390/fractalfract8080453 - 31 Jul 2024

Viewed by 793

Abstract

This paper presents an efficient finite difference method for solving the time-fractional Cattaneo equation with spatially variable coefficients in two spatial dimensions. The main idea is that the original equation is first transformed into a lower system, and then the graded mesh-based fast [...] Read more.

This paper presents an efficient finite difference method for solving the time-fractional Cattaneo equation with spatially variable coefficients in two spatial dimensions. The main idea is that the original equation is first transformed into a lower system, and then the graded mesh-based fast L2-

1_{σ}

formula and second-order spatial difference operator for the Caputo derivative and the spatial differential operator are applied, respectively, to derive the fully discrete finite difference scheme. By adding suitable perturbation terms, we construct an efficient fast second-order ADI finite difference scheme, which significantly improves computational efficiency for solving high-dimensional problems. The corresponding stability and error estimate are proved rigorously. Extensive numerical examples are shown to substantiate the accuracy and efficiency of the proposed numerical scheme. Full article

(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs)

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14 pages, 301 KiB

Open AccessArticle

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 1302

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

Open AccessArticle

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 1292

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 \leq 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 \leq 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

Open AccessArticle

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

Cited by 1 | Viewed by 1422

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

Open AccessArticle

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 1190

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

Open AccessArticle

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 39 | Viewed by 1611

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

Open AccessArticle

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 1205

Abstract

In this paper, we consider a fully discrete interpolated coefficient mixed finite element method for semilinear time fractional reaction–diffusion equations. The classic

L 1

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

L 1

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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