Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative

Çelik, Cem; Duman, Melda

doi:10.1016/j.jcp.2011.11.008

Cited by 436 publications

(228 citation statements)

References 14 publications

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“…where = ( + 1) 1−α − 1−α , = 0 1 · · · N. Secondly, we discretize the Riesz fractional derivative using the fractional centered difference scheme given in [34] …”

Section: Definitionmentioning

confidence: 99%

“…However, in order to better approximate the Riesz fractional derivative, Ortigueira [33] defined a 'fractional centered derivative' and proved that the Riesz fractional derivative of an analytic function can be represented by the fractional centered derivative. Celik and Duman [34] used the fractional centered derivative to approximate the Riesz fractional derivative and applied the Crank-Nicolson method to a fractional diffusion equation in the Riesz formulation, and showed that the method is unconditionally stable and convergent with accuracy two. In this paper, we use the fractional centered derivative to approximate the Riesz fractional derivative in Model-1 which can obtain second order accuracy in space, and propose an implicit numerical method.…”

Section: Introductionmentioning

confidence: 99%

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Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D

Liu

Turner

et al. 2013

Open Physics

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Abstract:Recently, the fractional Bloch-Torrey model has been used to study anomalous diffusion in the human brain.In this paper, we consider three types of space and time fractional Bloch-Torrey equations in two dimensions: Model-1 with the Riesz fractional derivative; Model-2 with the one-dimensional fractional Laplacian operator; and Model-3 with the two-dimensional fractional Laplacian operator. Firstly, we propose a spatially second-order accurate implicit numerical method for Model-1 whereby we discretize the Riesz fractional derivative using a fractional centered difference. We consider a finite domain where the time and space derivatives are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Secondly, we utilize the matrix transfer technique for solving Model-2 and Model-3. Finally, some numerical results are given to show the behaviours of these three models especially on varying domain sizes with zero Dirichlet boundary conditions.PACS (2008)

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“…where = ( + 1) 1−α − 1−α , = 0 1 · · · N. Secondly, we discretize the Riesz fractional derivative using the fractional centered difference scheme given in [34] …”

Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D

Liu

Turner

et al. 2013

Open Physics

View full text Add to dashboard Cite

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“…Compared with the tempered fractional PDEs, the finite difference methods for fractional PDEs have been much more well developed, e.g., [9,10,11,12]. A nature idea is to use the Grünwald-Letnikov formula [13] to approximate the Riemann-Liouville fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations

Deng

2017

Applied Numerical Mathematics

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Power-law probability density function (PDF) plays a key role in both subdiffusion and Lévy flights. However, sometimes because of the finite of the lifespan of the particles or the boundedness of the physical space, tempered power-law PDF seems to be a more physical choice and then the tempered fractional operators appear; in fact, the tempered fractional operators can also characterize the transitions among subdiffusion, normal diffusion, and Lévy flights. This paper focuses on the quasi-compact schemes for space tempered fractional diffusion equations, being much different from the ones for pure fractional derivatives. By using the generation function of the matrix and Weyl's theorem, the stability and convergence of the derived schemes are strictly proved. Some numerical simulations are performed to testify the effectiveness and numerical accuracy of the obtained schemes.

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“…Over the last decades, the finite difference methods have achieved some developments in solving the fractional differential equations, e.g., [6,14,19,34,36]. The Riemann-Liouville space fractional derivative can be naturally discretized by the standard Grünwald-Letnikov formula [23] with first order accuracy, but the finite difference schemes derived by the discretization are unconditionally unstable for the initial value problems including the implicit schemes that are well known to be stable most of the time for classical derivatives [19].…”

Section: Introductionmentioning

confidence: 99%

A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

Zhao

Deng

2014

Numer. Methods Partial Differential Eq.

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Based on the superconvergent approximation at some point (depending on the fractional order α, but not belonging to the mesh points) for Grünwald discretization to fractional derivative, we develop a series of high order quasi-compact schemes for space fractional diffusion equations. Because of the quasi-compactness of the derived schemes, no points beyond the domain are used for all the high order schemes including second order, third order, fourth order, and even higher order schemes; moreover, the algebraic equations for all the high order schemes have the completely same matrix structure. The stability and convergence analysis for some typical schemes are made; the techniques of treating the nonhomogeneous boundary conditions are introduced; and extensive numerical experiments are performed to confirm the theoretical analysis or verify the convergence orders.

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Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative

Cited by 436 publications

References 14 publications

Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D

Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D

Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations

A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

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