Numerical techniques for the variable order time fractional diffusion equation

Shen, Siqi; Liu, Fawang; Chen, Jing; Turner, Ian; Anh, Vo

doi:10.1016/j.amc.2012.04.047

Cited by 116 publications

(82 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many of these methods have since been generalised and extended in various ways, leading to non-standard finite difference schemes [10][11][12], finite difference schemes for problems of variable fractional order [13][14][15][16] and "fast" finite difference schemes [17][18][19][20][21] to name a few.…”

Section: Definition 2 (Riemann-liouville Fractional Derivatives On [ mentioning

confidence: 99%

See 1 more Smart Citation

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

2013

View full text Add to dashboard Cite

Abstract:We present a finite volume method to solve the time-space two-sided fractional advection-dispersion equation on a one-dimensional domain. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes. We demonstrate how the finite volume formulation provides a natural, convenient and accurate means of discretising this equation in conservative form, compared to using a conventional finite difference approach. Results of numerical experiments are presented to demonstrate the effectiveness of the approach. PACS

show abstract

Section: Definition 2 (Riemann-liouville Fractional Derivatives On [ mentioning

confidence: 99%

“…For sufficiently smooth functions C , the Riemann-Liouville derivatives (2), (3) and the Grünwald-Letnikov derivatives (15), (16) coincide [3, p.199]. The standard Grünwald formulas are obtained from (15) and (16) by replacing ∆ with the spatial step , yielding finite sum approximations.…”

Section: Definition 3 (Grünwald-letnikov Fractional Derivatives On [ mentioning

confidence: 99%

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

2013

View full text Add to dashboard Cite

show abstract

“…Lately, Shen et al [18] proposed numerical techniques for the variable-order time fractional diffusion equation. Zhang et al studied an implicit Euler numerical method for the time variable fractional-order mobile-immobile advection-dispersion model in [19].…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, the development of numerical methods to solve variable-order fractional differential equations is an actual and important problem. Nowadays numerical methods for variable-order fractional differential equations, which mainly cover finite difference methods [18][19][20][21][22][23][24][25][26], spectral methods [27][28][29], matrix methods [30,31], reproducing kernel methods [32,33], and so on, have been studied extensively by many researchers.…”

Section: Introductionmentioning

confidence: 99%

Finite difference scheme for multi-term variable-order fractional diffusion equation

Lü

Chen

et al. 2018

Adv Differ Equ

View full text Add to dashboard Cite

In this paper, we consider a multi-term variable-order fractional diffusion equation on a finite domain, which involves the Caputo variable-order time fractional derivative of order α(x, t) ∈ (0, 1) and the Riesz variable-order space fractional derivatives of order β(x, t) ∈ (0, 1), γ (x, t) ∈ (1, 2). Approximating the temporal direction derivative by L1-algorithm and the spatial direction derivative by the standard and shifted Grünwald method, respectively, a characteristic finite difference scheme is proposed. The stability and convergence of the difference schemes are analyzed via mathematical induction. Some numerical experiments are provided to show the efficiency of the proposed difference schemes.

show abstract

“…Fu et al [12] adopted the method of approximate particular solutions for both constant-order and variable-order time fractional diffusion models. Several finite difference methods for variable-order fractional partial diffusion equations were proposed in [13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Two New Approximations for Variable-Order Fractional Derivatives

Zhang

2017

Discrete Dynamics in Nature and Society

View full text Add to dashboard Cite

We introduced a parameter ( ) which was related to ( ); then two numerical schemes for variable-order Caputo fractional derivatives were derived; the second-order numerical approximation to variable-order fractional derivatives ( ) ∈ (0, 1) and 3 − ( )-order approximation for ( ) ∈ (1, 2) are established. For the given parameter ( ), the error estimations of formulas were proven, which were higher than some recently derived schemes. Finally, some numerical examples with exact solutions were studied to demonstrate the theoretical analysis and verify the efficiency of the proposed methods.

show abstract

Numerical techniques for the variable order time fractional diffusion equation

Cited by 116 publications

References 26 publications

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

Finite difference scheme for multi-term variable-order fractional diffusion equation

Two New Approximations for Variable-Order Fractional Derivatives

Contact Info

Product

Resources

About