Numerical Solution to the Multi-Term Time Fractional Diffusion Equation in a Finite Domain

Li, Gongsheng; Sun, Chunlong; Jia, Xin; Du, Di

doi:10.4208/nmtma.2016.y13024

Cited by 35 publications

(17 citation statements)

References 24 publications

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“…The above problem is motivated by the approximation of distributed order fractional models using a quadrature rule, see e.g. [4,7,20]. For the above multi-term models, by inserting the Hermite expansion (3.1) into the equation and imposing the collocation condition, one gets the following system:…”

Section: Applications To Multi-term Fractional Pdesmentioning

confidence: 99%

Hermite Spectral Collocation Methods for Fractional PDEs in Unbounded Domains

Tang¹,

Yuan²,

Zhou³

2018

CICP

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This work is concerned with spectral collocation methods for fractional PDEs in unbounded domains. The method consists of expanding the solution with proper global basis functions and imposing collocation conditions on the Gauss-Hermite points. In this work, two Hermite-type functions are employed to serve as basis functions. Our main task is to find corresponding differentiation matrices which are computed recursively. Two important issues relevant to condition numbers and scaling factors will be discussed. Applications of the spectral collocation methods to multi-term fractional PDEs are also presented. Several numerical examples are carried out to demonstrate the effectiveness of the proposed methods.

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Section: Applications To Multi-term Fractional Pdesmentioning

confidence: 99%

Hermite Spectral Collocation Methods for Fractional PDEs in Unbounded Domains

Tang¹,

Yuan²,

Zhou³

2018

CICP

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“…Moreover, thanks to the equality ∑ =1 + = 1 and with a similar method as used in [22], we get the unconditional stability and convergence of the difference scheme for any given finite time > 0.…”

Section: The Difference Scheme To the Forwardmentioning

confidence: 99%

“…For completeness of the paper, we give an implicit finite difference scheme in 1D case for solving the forward problem. For further details, see [22,23], and so on. …”

Section: The Forward Problem and The Inverse Problemmentioning

confidence: 99%

“…There are still a few research works reported on the multiterm TFDE like (1). On theoretical analysis and analytical methods for the forward problem, we refer to DaftardarGejji and Bhalekar [14], Luchko [15,16], Jiang et al [17], 2 Advances in Mathematical Physics Ding et al [18,19], and Li et al [20], and for numerical methods and simulations we refer to [21][22][23], and so on.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE

Sun¹,

Li²,

Jia³

2017

Advances in Mathematical Physics

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The fractional order in a fractional diffusion model is a key parameter which characterizes the anomalous diffusion behaviors. This paper deals with an inverse problem of determining the multiple fractional orders in the multiterm time-fractional diffusion equation (TFDE for short) from numerics. The homotopy regularization algorithm is applied to solve the inversion problem using the finite data at one interior point in the space domain. The inversion fractional orders with random noisy data give good approximations to the exact order demonstrating the efficiency of the inversion algorithm and numerical stability of the inversion problem.

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“…Zhang and Jiang studied a Crank-Nicolson Legendre spectral method [51] for nonlinear time fractional diffusion-wave equations. Since multi-term TFPDEs describe certain diffusion processes more accurately, various numerical methods, including Galerkin FEMs [3,17,50,56], orthogonal spline collocation methods [45], finite difference methods [5,16], compact difference methods [28], spectral methods [47,55], finite volume methods [46] have been developed for such equations. In addition, FEMs [35,36], compact finite difference methods [6,11], finite difference methods [38], Galerkin spectral element methods [6,30], singular boundary methods jointed with dual reciprocity methods [29] are also employed to multi-term TFWEs.…”

Section: Introductionmentioning

confidence: 99%

A Superconvergent Nonconforming Mixed FEM for Multi-Term Time-Fractional Mixed Diffusion and Diffusion-Wave Equations with Variable Coefficients

Fan¹

2021

EAJAM

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An unconditionally stable fully-discrete scheme on regular and anisotropic meshes for multi-term time-fractional mixed diffusion and diffusion-wave equations (TFMDDWEs) with variable coefficients is developed. The approach is based on a nonconforming mixed finite element method (FEM) in space and classical L1 time-stepping method combined with the Crank-Nicolson scheme in time. Then, the unconditionally stability analysis of the fully-discrete scheme is presented. The convergence for the original variable u and the flux p = µ(x)∇u, respectively, in H 1-and L 2-norms is derived by using the relationship between the projection operator R h and the interpolation operator I h. Interpolation postprocessing technique is used to establish superconvergence results. Finally, numerical tests are provided to demonstrate the theoretical analysis.

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Numerical Solution to the Multi-Term Time Fractional Diffusion Equation in a Finite Domain

Cited by 35 publications

References 24 publications

Hermite Spectral Collocation Methods for Fractional PDEs in Unbounded Domains

Hermite Spectral Collocation Methods for Fractional PDEs in Unbounded Domains

Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE

A Superconvergent Nonconforming Mixed FEM for Multi-Term Time-Fractional Mixed Diffusion and Diffusion-Wave Equations with Variable Coefficients

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