A High Order Finite Difference/Spectral Approximations to the Time Fractional Diffusion Equations

Cao, Jun; Xu, Chuan Ju; Wang, Zi Qiang

doi:10.4028/www.scientific.net/amr.875-877.781

Cited by 18 publications

(9 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Nowadays, there are already two types of second-order discretization schemes for Caputo-Fabrizio fractional operator: the first type is given in [8] based on the Fourier transform method and fractional linear multistep method [7,15]; and the second type is a L1 formula using linear interpolation approximation [14]. Based on the idea of [5,12,16], we employ the linear interpolation and quadratic interpolation approximation (L1−2 formula) to discrete the CF-fractional derivative, which derives a third-order discretization scheme.…”

Section: Time Discretizationmentioning

confidence: 99%

Finite difference/spectral approximations for the two-dimensional time Caputo-Fabrizio fractional diffusion equation

Yu,

Chen

2019

Preprint

View full text Add to dashboard Cite

The main contribution of this work is to construct and analyze stable and high order schemes to efficiently solve the two-dimensional time Caputo-Fabrizio fractional diffusion equation. Based on a third-order finite difference method in time and spectral methods in space, the proposed scheme is unconditionally stable and has the global truncation error O(τ 3 + N −m ), where τ , N and m are the time step size, polynomial degree and regularity in the space variable of the exact solution, respectively. It should be noted that the global truncation error O(τ 2 + N −m ) is well established in [ Li, Lv and Xu, Numer. Methods Partial Differ. Equ. (2019)]. Finally, some numerical experiments are carried out to verify the theoretical analysis. To the best of our knowledge, this is the first proof for the stability of the third-order scheme for the Caputo-Fabrizio fractional operator.

show abstract

Section: Time Discretizationmentioning

confidence: 99%

Finite difference/spectral approximations for the two-dimensional time Caputo-Fabrizio fractional diffusion equation

Yu,

Chen

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Theorem 1. Let w be the exact solution of (1), (2), 0 { } k K k w  be the numerical solution of (8). Then the following error estimates hold…”

Section: The Convergence Analysismentioning

confidence: 99%

“…Based on the so-called block-by-block approach, Cao and Xu constructed a high order schema for fractional differential equations of the order [7]. Based on a finite difference scheme in time and Legendre spectral methods in space, Cao, Xu and Wang constructed high order scheme to efficiently solve the time-fractional diffusion equation [8].…”

Section: Introductionmentioning

confidence: 99%

A High Order Numerical Algorithm for the Model of Viscoelastic Fractional Derivative

Wang¹,

Yang²,

Cao³

2017

dtcse

View full text Add to dashboard Cite

In this paper, we construct a high order scheme to efficiently solve the forced vibration equation of viscoelastic material. The proposed method is based on a finite difference scheme in time.We used central difference scheme to the second derivative with second order accurate and 3   order accurate scheme to the fractional derivative of the order ,0 1     . We obtain that the numerical scheme is second order. The convergence analysis is given that the numerical approximation of the exact solution accuracy as a second order. A series of numerical examples are given to verify the correctness of the theoretical analysis.

show abstract

“…In fact, the L1 formula is established by a piecewise linear interpolation approximation for the integrand function on each small interval. By applying a higher-order interpolant instead of the linear interpolant to improve the numerical accuracy, Cao et al [5], Gao et al [12], and Lv and Xu [20] proposed some (3 − )-order formulae for the -order Caputo fractional derivative. A three-point L1 approximation of the Caputo derivative with order 3 − has been also presented in [11].…”

Section: Introductionmentioning

confidence: 99%

A High Order Formula to Approximate the Caputo Fractional Derivative

Mokhtari

Mostajeran

2019

Commun. Appl. Math. Comput.

View full text Add to dashboard Cite

We present here a high-order numerical formula for approximating the Caputo fractional derivative of order for 0 < < 1. This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial differential equations. In comparison with the previous formulae, the main superiority of the new formula is its order of accuracy which is 4 − , while the order of accuracy of the previous ones is less than 3. It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost. The effectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples. Moreover, an application of the new formula in solving some fractional partial differential equations is presented by constructing a finite difference scheme. A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.

show abstract

A High Order Finite Difference/Spectral Approximations to the Time Fractional Diffusion Equations

Cited by 18 publications

References 16 publications

Finite difference/spectral approximations for the two-dimensional time Caputo-Fabrizio fractional diffusion equation

Finite difference/spectral approximations for the two-dimensional time Caputo-Fabrizio fractional diffusion equation

A High Order Numerical Algorithm for the Model of Viscoelastic Fractional Derivative

A High Order Formula to Approximate the Caputo Fractional Derivative

Contact Info

Product

Resources

About