Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

Yang, Qianqian; Liu, Fawang; Turner, Ian

doi:10.1016/j.apm.2009.04.006

Cited by 523 publications

(253 citation statements)

References 31 publications

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“…where Using the relationship between the Riemann-Liouville derivative and the Grünwald-Letnikov scheme, we discretize the Riesz fractional derivative by the shifted Grünwald-Letnikov scheme in Yang et al [24] 6) where the coefficients are defined by…”

Section: Implicit Numerical Methods For the St-fbtementioning

confidence: 99%

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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.

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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.

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Section: Implicit Numerical Methods For the St-fbtementioning

confidence: 99%

“…We note in passing that the Riesz and fractional Laplacian formulations are not equivalent, except for certain special cases. For example, in one dimension with zero Dirichlet boundary conditions, these two formulations are the same [24].…”

Section: Introductionmentioning

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.

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“…The non-integer order spatial derivative was given by Riesz and has the following form (see, e.g., Yang et al, 2010).…”

Section: Definition 5 the Riemann-liouville Definition Of The Fo Opementioning

confidence: 99%

Modeling heat distribution with the use of a non-integer order, state space model

Oprzźdkiewicz

Gawin

Mitkowski

2016

International Journal of Applied Mathematics and Computer Science

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A new, state space, non-integer order model for the heat transfer process is presented. The proposed model is based on a Feller semigroup one, the derivative with respect to time is expressed by the non-integer order Caputo operator, and the derivative with respect to length is described by the non-integer order Riesz operator. Elementary properties of the state operator are proven and a formula for the step response of the system is also given. The proposed model is applied to the modeling of temperature distribution in a one dimensional plant. Results of experiments show that the proposed model is more accurate than the analogical integer order model in the sense of the MSE cost function.

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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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Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

Cited by 523 publications

References 31 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

Modeling heat distribution with the use of a non-integer order, state space model

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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