Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection-diffusion equation

Shen, Siqi; Liu, Fawang; Anh, Vo

doi:10.1007/s11075-010-9393-x

Cited by 132 publications

(66 citation statements)

References 25 publications

(31 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…At a point (x i , y j , z k ) at the moment of time t n for i, j, k ∈ N and n ∈ N , we denote the exact and numerical solutions M(r, t) as u(x i , y j , z k , t n ) and u n i,j,k , respectively. Firstly, adopting the discrete scheme in Shen et al [28], we discretize the Caputo time fractional derivative of u(x i , y j , z k , t n+1 ) as…”

Section: Implicit Numerical Methods For the St-fbtementioning

confidence: 99%

Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation

Liu

Turner

et al. 2013

Phil. Trans. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

Fractional-order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brownian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As magnetic resonance imaging is applied with increasing temporal and spatial resolution, the spin dynamics is being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here, the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments, where processes are often anisotropic. Anomalous diffusion in the human brain using fractional-order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch–Torrey equation using fractional-order calculus with respect to time and space. However, effective numerical methods and supporting error analyses for the fractional Bloch–Torrey equation are still limited. In this paper, the space and time fractional Bloch–Torrey equation (ST-FBTE) in both fractional Laplacian and Riesz derivative form is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE in fractional Laplacian form with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE based on the Riesz form, and the stability and convergence of the INM are investigated. We prove that the INM for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

show abstract

Section: Implicit Numerical Methods For the St-fbtementioning

confidence: 99%

Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation

Liu

Turner

et al. 2013

Phil. Trans. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Yu et al [22] proposed a fractional alternating direction implicit scheme to overcome this problem, they also proved the stability and convergence of the proposed method with order of convergence one in space. For the Riesz fractional formulation, the Gr¨ nwaldLetnikov derivative approximation scheme of order one can be used [22,23,[30][31][32]. 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.…”

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

Self Cite

View full text Add to dashboard Cite

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)

show abstract

“…The recent works can see the references [17][18][19]. However, high order accuracy schemes are seldom derived by finite difference method.…”

Section: Introductionmentioning

confidence: 99%

Pseudo-Spectral Method for Space Fractional Diffusion Equation

Huang¹,

Zheng²

2013

View full text Add to dashboard Cite

This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogonal polynomials or interpolation polynomials. Then, by using pseudo-spectral method, the SFDE is reduced to a system of ordinary differential equations for time variable t. The high order Runge-Kutta scheme can be used to solve the system. So, a high order numerical scheme is derived. Numerical examples illustrate that the results obtained by this method agree well with the analytical solutions.

show abstract

Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection-diffusion equation

Cited by 132 publications

References 25 publications

Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation

Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation

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

Pseudo-Spectral Method for Space Fractional Diffusion Equation

Contact Info

Product

Resources

About