A Legendre spectral method on graded meshes for the two-dimensional multi-term time-fractional diffusion equation with non-smooth solutions

Zheng, Rumeng; Liu, Fawang; Jiang, Xiaoyun

doi:10.1016/j.aml.2020.106247

Cited by 19 publications

(7 citation statements)

References 21 publications

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“…Many authors have considered various numerical methods for the multi-term fractional subdiffusion problems [4][5][6][7][8][9], where numerical discrete schemes concentrate on stability and high order convergence. The study of some nice properties, such as positivity preservation, local conservation, discrete extremum principle, is much less considered and developed.…”

Section: Introductionmentioning

confidence: 99%

Simple positivity-preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes

Yang

Zhang

et al. 2022

Nonlinear Dyn

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Section: Introductionmentioning

confidence: 99%

Simple positivity-preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes

Yang

Zhang

et al. 2022

Nonlinear Dyn

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“…Parallelly, there are extensive profound literatures concerning theory and computation of the forward problems for the multi-term time-fractional diffusion equations (see [15,24,26,29,32,52] for an incomplete list). On the other hand, the published works were considered from different aspect on the inverse problems for multi-term counterpart, e.g., see [11-13, 23, 25, 27, 28, 43, 44, 47] etc.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous inversion of the potential term and the fractional orders in a multi-term time-fractional diffusion equation

Sun¹,

Li²,

Zhang³

2021

Inverse Problems

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In the present paper, we devote our effort to a nonlinear inverse problem for simultaneously recovering the potential function and the fractional orders in a multi-term time-fractional diffusion equation from the noisy boundary Cauchy data in the one-dimensional case. The uniqueness for the inverse problem is derived based on the analytic continuation, the Laplace transformation and the Gel’fand–Levitan theory. Finally, the Levenberg–Marquardt regularization method with a regularization parameter chosen by a sigmoid-type function is applied for finding a stable approximate solution. Three numerical examples are provided to show the effectiveness of the proposed method.

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“…Notice that the DST technology can avoid solving matrix inversion directly and has been successfully applied in the discretization of classical models, such as Poisson equation [11] and general order semilinear evolution equations [12], just to name a few. On the other hand, the weak singularity of the fractional model has gradually attracted the attention of scholars in the fractional community, and some kinds of methods have been proposed to resolve this issue, such as nonuniform meshes [13][14][15][16] and convolution quadrature [9,17,18]. The method of adding correction terms is also an efficient way of dealing with nonsmooth solutions problems.…”

Section: Introductionmentioning

confidence: 99%

Fast High-Order Difference Scheme for the Modified Anomalous Subdiffusion Equation Based on Fast Discrete Sine Transform

Nong

Chen

2021

Journal of Function Spaces

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The modified anomalous subdiffusion equation plays an important role in the modeling of the processes that become less anomalous as time evolves. In this paper, we consider the efficient difference scheme for solving such time-fractional equation in two space dimensions. By using the modified L1 method and the compact difference operator with fast discrete sine transform technique, we develop a fast Crank-Nicolson compact difference scheme which is proved to be stable with the accuracy of O τ min 1 + α , 1 + β + h 4 . Here, α and β are the fractional orders which both range from 0 to 1, and τ and h are, respectively, the temporal and spatial stepsizes. We also consider the method of adding correction terms to efficiently deal with the nonsmooth problems. Numerical examples are provided to verify the effectiveness of the proposed scheme.

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A Legendre spectral method on graded meshes for the two-dimensional multi-term time-fractional diffusion equation with non-smooth solutions

Cited by 19 publications

References 21 publications

Simple positivity-preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes

Simple positivity-preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes

Simultaneous inversion of the potential term and the fractional orders in a multi-term time-fractional diffusion equation

Fast High-Order Difference Scheme for the Modified Anomalous Subdiffusion Equation Based on Fast Discrete Sine Transform

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