High-order algorithms for Riesz derivative and their applications (II)

Ding, Hengfei; Li, Changpin; Chen, YangQuan

doi:10.1016/j.jcp.2014.06.007

Cited by 119 publications

(57 citation statements)

References 30 publications

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“…Its analytical solution is u(x, t) = exp(t)x 6 (1 − x) 6 and satisfy the corresponding initial and boundary values conditions. We solve this problem with the numerical schemes (3.6) and (3.11) for different values of τ , h and α.…”

Section: Example 2 Consider the Following Equationmentioning

confidence: 99%

“…For more details, see the recent publications [1,2,3,5,6,7,8,9,10,14,15,17,19,20,21,22], and references cited therein. The Riemann-Liouville (R-L) derivative and Caputo derivative are commonly used, respectively defined below: The left R-L derivative reflects the dependence on the history, while the right R-L derivative the dependence upon the future.…”

Section: Introductionmentioning

confidence: 99%

“…Shortly after, some other high-order algorithms for Caputo derivatives were also appeared, for example see [1,3,14] and references cited therein. And the high-order algorithms for Riesz derivatives were derived in [5,6]. In [5], Ding et al constructed the fourth-order schemes for the Riesz derivative and applied them to the space Riesz fractional diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

“…In that paper they established the fourth-order schemes for R-L derivatives [5] from which some similar fourth-order schemes were derived by other people. In [6], Ding et al continued to establish the sixth-order, eighth-order, tenth-order and twelfth-order schemes for the Riesz derivative by using the Fourier analysis, where the fourth-order compact scheme for R-L derivative was especially highlighted. Then they used the two schemes, among them to the Riesz space fractional reaction-dispersion equation, where the rigorous error analysis was given.…”

Section: Introductionmentioning

confidence: 99%

“…Then they used the two schemes, among them to the Riesz space fractional reaction-dispersion equation, where the rigorous error analysis was given. The other odd-order (third-order, fifth-order, seventh-order, ninth-order, eleventh-order) schemes can be established from [6] by choosing the different parameters therein. In spite of these, it is absolutely necessary to construct intuitionist and straightforward schemes for the Riesz derivatives.…”

Section: Introductionmentioning

confidence: 99%

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High-Order Algorithms for Riesz Derivative and their Applications (III)

Ding

2016

FCAA

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Numerical methods for fractional calculus attract increasing interest due to its wide applications in various fields such as physics, mechanics, etc. In this paper, we focus on constructing high-order algorithms for Riesz derivatives, where the convergence orders cover from the second order to the sixth order. Then we apply the established schemes to the Riesz type turbulent diffusion equation (or, Riesz space fractional turbulent diffusion equation). Numerical experiments are displayed which support the theoretical analysis.MSC 2010 : 65M06, 65M12

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Section: Example 2 Consider the Following Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High-Order Algorithms for Riesz Derivative and their Applications (III)

Ding

2016

FCAA

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A space‐time spectral collocation method for the 2‐dimensional nonlinear Riesz space fractional diffusion equations

Jiang

2018

Math Methods in App Sciences

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The space-time spectral collocation method was initially presented for the 1-dimensional sine-Gordon equation. In this article, we introduce a space-time spectral collocation method for solving the 2-dimensional nonlinear Riesz space fractional diffusion equations. The method is based on a Legendre-Gauss-Lobatto spectral collocation method for discretizing spatial and the spectral collocation method for the time nonlinear first-order system of ordinary differential equation. Optimal priori error estimates in L 2 norms for the semidiscrete formulation and the uniqueness of the approximate solution are derived. The method has spectral accuracy in both space and time, and the numerical results confirm the statement. KEYWORDSerror estimates, nonlinear initial-value problems, Riesz fractional derivative, space-time spectral method, spectral collocation method 6130

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Spectral collocation method for a class of fractional diffusion differential equations with nonsmooth solutions

Wang

Xiao

2020

Math Methods in App Sciences

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In this paper, we develop spectral collocation method for a class of fractional diffusion differential equations. Since the solutions of these fractional differential equations usually exhibit singularities at the end‐points, it can not be well approximated by classical polynomial basis functions. We first give nonclassical interpolants based on the associated Jacobi‐Gauss points and obtain the corresponding fractional differentiation matrices. Second, we prove the convergence for the developed spectral collocation method. Finally, several numerical examples are considered to demonstrate the validity and applicability of the developed basis functions.

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High-order algorithms for Riesz derivative and their applications (II)

Cited by 119 publications

References 30 publications

High-Order Algorithms for Riesz Derivative and their Applications (III)

High-Order Algorithms for Riesz Derivative and their Applications (III)

A space‐time spectral collocation method for the 2‐dimensional nonlinear Riesz space fractional diffusion equations

Spectral collocation method for a class of fractional diffusion differential equations with nonsmooth solutions

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