Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 

Saved Queries

Sign in to use this feature.

Search Filter Reset All

Years

Between: -

Subjects

remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
Add Subjects Reset
Select Subjects

Journals

remove_circle_outline
remove_circle_outline
remove_circle_outline
Add Journals Reset
Select Journals

Article Types

remove_circle_outline
Add Article Types Reset
Select Article Types

Countries / Regions

remove_circle_outline
remove_circle_outline
Add Countries / Regions Reset
Select Countries / Regions
Update Search
share announcement

Search Results (5)

Search Parameters:
Keywords = time-fractional diffusion equation (TFDE)

Order results
Result details
Results per page
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
28 pages, 1721 KB  
Article
Stability and Convergence Analysis of Compact Finite Difference Method for High-Dimensional Time-Fractional Diffusion Equations with High-Order Accuracy in Time
by Jun-Ying Cao, Jian-Qiang Fang, Zhong-Qing Wang and Zi-Qiang Wang
Fractal Fract. 2025, 9(8), 520; https://doi.org/10.3390/fractalfract9080520 - 8 Aug 2025
Viewed by 377
Abstract
Based on the spatial compact finite difference (SCFD) method, an improved high-order temporal accuracy scheme for high-dimensional time-fractional diffusion equations (TFDEs) is presented in this work. Combining the temporal piecewise quadratic interpolation and the high-dimensional SCFD method, the proposed numerical method is described. [...] Read more.
Based on the spatial compact finite difference (SCFD) method, an improved high-order temporal accuracy scheme for high-dimensional time-fractional diffusion equations (TFDEs) is presented in this work. Combining the temporal piecewise quadratic interpolation and the high-dimensional SCFD method, the proposed numerical method is described. In order to establish the stability and convergence analysis, we introduce a norm ||·||H˜1, which is rigorously proved equivalent to the standard H1-norm. Considering that the coefficients of high-order numerical schemes are not entirely positive, we introduce an appropriate parameter to transform the numerical scheme into an equivalent form with positive coefficients. Based on the equivalent form, we prove that the temporal and spatial convergence orders are (3−γ) and 4 by applying the convergence of geometric progression. The proposed scheme ensures that the theoretical convergence accuracy at each time step is of order (3−γ) without requiring any additional processing techniques. Ultimately, the convergence of the proposed high-order accurate scheme is verified through numerical experiments involving (non-)linear high-dimensional TFDEs. Full article
(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs, Second Edition)
►▼ Show Figures

Figure 1

21 pages, 476 KB  
Article
A New L2 Type Difference Scheme for the Time-Fractional Diffusion Equation
by Cheng-Yu Hu and Fu-Rong Lin
Fractal Fract. 2025, 9(5), 325; https://doi.org/10.3390/fractalfract9050325 - 20 May 2025
Viewed by 612
Abstract
In this paper, a new L2 (NL2) scheme is proposed to approximate the Caputo temporal fractional derivative, leading to a time-stepping scheme for the time-fractional diffusion equation (TFDE). Subsequently, the space derivative of the resulting system is discretized using a specific finite difference [...] Read more.
In this paper, a new L2 (NL2) scheme is proposed to approximate the Caputo temporal fractional derivative, leading to a time-stepping scheme for the time-fractional diffusion equation (TFDE). Subsequently, the space derivative of the resulting system is discretized using a specific finite difference method, yielding a fully discrete system. We then establish the H1-norm stability and convergence of the time-stepping scheme on uniform meshes for the TFDE. In particular, we prove that the proposed scheme has (3−α)th-order accuracy, where α (0<α<1) is the order of the time-fractional derivative. Finally, numerical experiments for several test problems are carried out to validate the obtained theoretical results. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)
►▼ Show Figures

Figure 1

15 pages, 1679 KB  
Article
A Highly Accurate Computational Approach to Solving the Diffusion Equation of a Fractional Order
by Haifa Bin Jebreen
Mathematics 2024, 12(13), 1965; https://doi.org/10.3390/math12131965 - 25 Jun 2024
Viewed by 1112
Abstract
This study aims to present and apply an effective algorithm for solving the TFDE (Time-Fractional Diffusion Equation). The Chebyshev cardinal polynomials and the operational matrix for fractional derivatives based on these bases are relied on as crucial tools to achieve this objective. By [...] Read more.
This study aims to present and apply an effective algorithm for solving the TFDE (Time-Fractional Diffusion Equation). The Chebyshev cardinal polynomials and the operational matrix for fractional derivatives based on these bases are relied on as crucial tools to achieve this objective. By employing the pseudospectral method, the equation is transformed into an algebraic linear system. Consequently, solving this system using the GMRES method (Generalized Minimal Residual) results in obtaining the solution to the TFDE. The results obtained are very accurate, and in certain instances, the exact solution is achieved. By solving some numerical examples, the proposed method is shown to be effective and yield superior outcomes compared to existing methods for addressing this problem. Full article
(This article belongs to the Special Issue Advances in Fractional Operators and Their Applications in Physical Sciences)
►▼ Show Figures

Figure 1

20 pages, 1277 KB  
Article
A Preconditioner for Galerkin–Legendre Spectral All-at-Once System from Time-Space Fractional Diffusion Equation
by Meijuan Wang and Shugong Zhang
Symmetry 2023, 15(12), 2144; https://doi.org/10.3390/sym15122144 - 2 Dec 2023
Viewed by 1350
Abstract
As a model that possesses both the potentialities of Caputo time fractional diffusion equation (Caputo-TFDE) and symmetric two-sided space fractional diffusion equation (Riesz-SFDE), time-space fractional diffusion equation (TSFDE) is widely applied in scientific and engineering fields to model anomalous diffusion phenomena including subdiffusion [...] Read more.
As a model that possesses both the potentialities of Caputo time fractional diffusion equation (Caputo-TFDE) and symmetric two-sided space fractional diffusion equation (Riesz-SFDE), time-space fractional diffusion equation (TSFDE) is widely applied in scientific and engineering fields to model anomalous diffusion phenomena including subdiffusion and superdiffusion. Due to the fact that fractional operators act on both temporal and spatial derivative terms in TSFDE, efficient solving for TSFDE is important, where the key is solving the corresponding discrete system efficiently. In this paper, we derive a Galerkin–Legendre spectral all-at-once system from the TSFDE, and then we develop a preconditioner to solve this system. Symmetry property of the coefficient matrix in this all-at-once system is destroyed so that the deduced all-at-once system is more convenient for parallel computing than the traditional timing-step scheme, and the proposed preconditioner can efficiently solve the corresponding all-at-once system from TSFDE with nonsmooth solution. Moreover, some relevant theoretical analyses are provided, and several numerical results are presented to show competitiveness of the proposed method. Full article
(This article belongs to the Special Issue New Challenges in Algorithms/Design/Process Optimization with Symmetry/Asymmetry)
►▼ Show Figures

Figure 1

11 pages, 286 KB  
Article
Analysis of a Hidden-Memory Variably Distributed-Order Time-Fractional Diffusion Equation
by Jinhong Jia
Fractal Fract. 2022, 6(11), 627; https://doi.org/10.3390/fractalfract6110627 - 28 Oct 2022
Cited by 1 | Viewed by 1563
Abstract
We analyze the well-posedness and regularity of a variably distributed-order time-fractional diffusion equation (tFDE) with a hidden-memory fractional derivative, which provide a competitive means to describe the anomalously diffusive transport of particles in heterogeneous media. We prove that the solution of a variably [...] Read more.
We analyze the well-posedness and regularity of a variably distributed-order time-fractional diffusion equation (tFDE) with a hidden-memory fractional derivative, which provide a competitive means to describe the anomalously diffusive transport of particles in heterogeneous media. We prove that the solution of a variably distributed-order tFDE has weak singularity at the initial time t=0 which depends on the upper bound of a distributed order α¯(0). Full article
(This article belongs to the Special Issue Variable-Order Fractional Problems: Modeling, Analysis, Approximation and Application)
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying article 1-50 on page 1 of 1.
Back to TopTop