Fast implicit difference schemes for time‐space fractional diffusion equations with the integral fractional Laplacian

Gu, Xian Ming; Sun, Hai-Wei; Zhang, Yanzhi; Zhao, Yong Liang

doi:10.1002/mma.6746

Cited by 20 publications

(10 citation statements)

References 61 publications

(208 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 5. In [26,27], authors studied the time-space fractional evolution equations with nonlocal Laplacian by difference methods. We obtain the mild solution by the technology of eigenvalues' expansions about nonlocal Laplacian operators.…”

Section: Remarkmentioning

confidence: 99%

See 1 more Smart Citation

Three-Point Boundary Value Problems of Coupled Nonlocal Laplacian Equations

Zhou

Zhang

2022

Mathematics

View full text Add to dashboard Cite

This paper discusses a kind of coupled nonlocal Laplacian evolution equation with Caputo time-fractional derivatives and proportional delays. Green function and mild solution are firstly established by employing the approach of eigenvalues’ expansions and Fourier analysis technique. By the properties of eigenvalues and Mittag–Leffler functions, several vital estimations of Green functions are presented. In view of these estimations and some appropriate assumptions, the existence and uniqueness of the mild solution are studied by utilizing the Leray–Schauder fixed-point theorem and the Banach fixed-point theorem. Finally, an example is provided to illustrate the effectiveness of our main results.

show abstract

Section: Remarkmentioning

confidence: 99%

“…where c D α 0 represents Caputo derivative with order α ∈ (0, 1), (−∆) s is the nonlocal Laplacian operator with order s ∈ (0, 1). Additional content regarding this theme is available in [27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Three-Point Boundary Value Problems of Coupled Nonlocal Laplacian Equations

Zhou

Zhang

2022

Mathematics

View full text Add to dashboard Cite

show abstract

“…Thus, we need to obtain u 1 i first by other methods. In this work, we use the following fast L1 scheme [48,49] to get u 1 i :…”

Section: The L2-type Difference Scheme and Its Stabilitymentioning

confidence: 99%

On the bilateral preconditioning for an L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations

Zhao¹,

Gu³

2021

Preprint

View full text Add to dashboard Cite

Time-space fractional Bloch-Torrey equations are developed by some researchers to investigate the relationship between diffusion and fractional-order dynamics. In this paper, we first propose a second-order scheme for this equation by employing the recently proposed L2-type formula [A. A. Alikhanov, C. Huang, Appl. Math. Comput. ( 2021) 126545]. Then, we prove the stability and the convergence of this scheme.Based on such the numerical scheme, a L2-type all-at-once system is derived. In order to solve this system in a parallel-in-time pattern, a bilateral preconditioning technique is designed according to the special structure of the system. We theoretically show that the condition number of the preconditioned matrix is uniformly bounded by a constant for the time fractional order α ∈ (0, 0.3624). Numerical results are reported to show the efficiency of our method.

show abstract

“…[20] proposed a linear quasi-compact finite difference scheme for semi-linear space-fractional diffusion equations with time delays. And the time-space fractional nonlinear diffusion equation received attention in [21,22]. Furthermore, Refs.…”

Section: Introductionmentioning

confidence: 99%

Optimal H1-Norm Estimation of Nonconforming FEM for Time-Fractional Diffusion Equation on Anisotropic Meshes

et al. 2022

View full text Add to dashboard Cite

In this paper, based on the L2-1σ scheme and nonconforming EQ1rot finite element method (FEM), a numerical approximation is presented for a class of two-dimensional time-fractional diffusion equations involving variable coefficients. A novel and detailed analysis of the equations with an initial singularity is described on anisotropic meshes. The fully discrete scheme is shown to be unconditionally stable, and optimal second-order accuracy for convergence and superconvergence can be achieved in both time and space directions. Finally, the obtained numerical results are compared with the theoretical analysis, which verifies the accuracy of the proposed method.

show abstract

Fast implicit difference schemes for time‐space fractional diffusion equations with the integral fractional Laplacian

Abstract: In recent decades, fractional partial differential equations (FPDEs) have attracted growing attention in modeling phenomena with long-term memory and spatial heterogeneity arising in engineering, physics, chemistry, and other applied

Cited by 20 publications

References 61 publications

Three-Point Boundary Value Problems of Coupled Nonlocal Laplacian Equations

Three-Point Boundary Value Problems of Coupled Nonlocal Laplacian Equations

On the bilateral preconditioning for an L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations

Optimal H1-Norm Estimation of Nonconforming FEM for Time-Fractional Diffusion Equation on Anisotropic Meshes

Contact Info

Product

Resources

About