An implicit numerical method for the two-dimensional fractional percolation equation

Chen, S.; Liu, F.; Turner, Ian; Anh, Vo

doi:10.1016/j.amc.2012.10.003

Cited by 37 publications

(28 citation statements)

References 16 publications

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“…To prove the stability and convergence of the numerical method, we need the following lemma in [24].…”

Section: Stability and Convergence Of The Adimmentioning

confidence: 99%

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Liu¹,

Chen²,

Turner³

et al. 2013

Open Physics

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Abstract:Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.PACS (2008)

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“…To prove the stability and convergence of the numerical method, we need the following lemma in [24].…”

Section: Stability and Convergence Of The Adimmentioning

confidence: 99%

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Liu¹,

Chen²,

Turner³

et al. 2013

Open Physics

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show abstract

“…But these fractional differential equations are difficult to get their exact solutions [9,12,13,21]. So, these types of equations are solved by various methods such as Adomian decomposition method [1,22], variational iteration method [7,20], differential transform method [2,6], homotopy perturbation method [11,15], an iterative method [4,23], finite element method [19,25], finite difference method [3,18], etc..…”

Section: Introductionmentioning

confidence: 99%

A new Mittag-Leffler function undetermined coefficient method and its applications to fractional homogeneous partial differential equations

Li¹,

Sun²,

Yin³

et al. 2017

J. Nonlinear Sci. Appl.

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In this paper, we develop a new application of the Mittag-Leffler function that will extend the application to fractional homogeneous differential equations, and propose a Mittag-Leffler function undetermined coefficient method. A new solution is constructed in power series. When a very simple ordinary differential equation is satisfied, no matter the original equation is linear or nonlinear, the method is valid, then combine the alike terms, compare the coefficient with identical powers, and the undetermined coefficient will be obtained. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided, and the solutions are in the form of generalized Mittag-Leffler function. The results reveal that the approach introduced here are very effective and convenient for solving homogeneous differential equations with fractional order.

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“…Eq. (2.3) may be rearranged and written as the fractional Crank-Nicholson discretization in the form 4) and the operator form (2.4) may be written in matrix form.…”

Section: The Crank-nicholson Methods For the One-dimensional Fpementioning

confidence: 99%

“…Guo et al [6] proposed an implicit finite difference method for the one-dimensional fractional percolation equation with Dirichlet and fractional boundary conditions. Chen et al [4] considered an alternating direction implicit difference method for the two-dimensional case. Chen et al [3] discussed the two-dimensional variable-order fractional percolation equation.…”

Section: Introductionmentioning

confidence: 99%

Second-order accurate numerical approximations for the fractional percolation equations

Yin¹,

Li²,

Fang³

2017

J. Nonlinear Sci. Appl.

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First, we examine a practical numerical method which based on the classical Crank-Nicholson (CN) method combined with Richardson extrapolation is used to solve a class of one-dimensional initial-boundary value fractional percolation equation (FPE) with variable coefficients on a finite domain. Secondly, we present ADI-CN method for the two-dimensional fractional percolation equation. Stability and convergence of these methods are proved. Using these methods, we can achieve second-order convergence in time and space. Finally, numerical examples are presented to verify the order of convergence.

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An implicit numerical method for the two-dimensional fractional percolation equation

Cited by 37 publications

References 16 publications

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

A new Mittag-Leffler function undetermined coefficient method and its applications to fractional homogeneous partial differential equations

Second-order accurate numerical approximations for the fractional percolation equations

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