Product rule for vector fractional derivatives

Bolster, Diogo; Meerschaert, Mark M.; Sikorskii, Alla

doi:10.2478/s13540-012-0033-0

Cited by 6 publications

(7 citation statements)

References 17 publications

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“…1 shows the response of the space fractional diffusion Eq. (14) with boundary and initial conditions (15,16) using the proposed full-discrete scheme (9) for different diffusion coefficients j. It indicates that the solution decays more quickly while the diffusion coefficient j increases.…”

Section: Examplementioning

confidence: 95%

“…They proved the existence and uniqueness of the solution of the finite difference system. Bolster et al proved a product rule for vector fractional derivatives using Fourier transforms [14]. Currently, the FDM is one of dominant numerical methods for solving FPDEs and some high-order finite difference methods for fractional partial differential equations attract increasing interests [15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

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A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation

Liu

et al. 2015

Applied Mathematics and Computation

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a b s t r a c t This paper aims to develop a meshless approach based on the Point Interpolation Method (PIM) for numerical simulation of a space fractional diffusion equation. Two fully-discrete schemes for the one-dimensional space fractional diffusion equation are obtained by using the PIM and the strong-forms of the space diffusion equation. Numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach.

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Section: Examplementioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation

Liu

et al. 2015

Applied Mathematics and Computation

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“…The nonlinear nature of r can be seen using the rules of fractional vector calculus outlined in [9, 36, 32]. Reaction rate r ( x , t ) can be written as (see appendix A for details)…”

Section: Spatial Fractional Advection Dispersion Reaction Systemmentioning

confidence: 99%

“…In this paper we focus exclusively on the role of spatial nonlocality and extend the work of [7] to a system with transport governed by a multi-dimensional space fADE. We do so by combining and extending the fractional calculus methods developed in [9, 36, 32]. …”

Section: Introductionmentioning

confidence: 99%

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Mixing-driven equilibrium reactions in multidimensional fractional advection–dispersion systems

Bolster

Benson

Meerschaert

et al. 2013

Physica A: Statistical Mechanics and its Applications

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We study instantaneous, mixing-driven, bimolecular equilibrium reactions in a system where transport is governed by a multidimensional space fractional dispersion equation. The superdiffusive, nonlocal nature of the system causes the location and magnitude of reactions that take place to change significantly from a classical Fickian diffusion model. In particular, regions where reaction rates would be zero for the Fickian case become regions where the maximum reaction rate occurs when anomalous dispersion operates. We also study a global metric of mixing in the system, the scalar dissipation rate and compute its asymptotic scaling rates analytically. The scalar dissipation rate scales asymptotically as t−(d+α)/α, where d is the number of spatial dimensions and α is the fractional derivative exponent.

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Determination of Coefficients of High-Order Schemes for Riemann-Liouville Derivative

Ding

2014

The Scientific World Journal

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Although there have existed some numerical algorithms for the fractional differential equations, developing high-order methods (i.e., with convergence order greater than or equal to 2) is just the beginning. Lubich has ever proposed the high-order schemes when he studied the fractional linear multistep methods, where he constructed the pth order schemes (p = 2, 3, 4, 5, 6) for the αth order Riemann-Liouville integral and αth order Riemann-Liouville derivative. In this paper, we study such a problem and develop recursion formulas to compute these coefficients in the higher-order schemes. The coefficients of higher-order schemes (p = 7,8, 9,10) are also obtained. We first find that these coefficients are oscillatory, which is similar to Runge's phenomenon. So, they are not suitable for numerical calculations. Finally, several numerical examples are implemented to testify the efficiency of the numerical schemes for p = 3,…, 6.

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Product rule for vector fractional derivatives

Abstract: This paper establishes a product rule for fractional derivatives of a realvalued function defined on a finite dimensional Euclidean vector space. The proof uses Fourier transforms.MSC 2010 : Primary: 26A33, 26B12; Secondary: 60E07

Cited by 6 publications

References 17 publications

A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation

A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation

Mixing-driven equilibrium reactions in multidimensional fractional advection–dispersion systems

Determination of Coefficients of High-Order Schemes for Riemann-Liouville Derivative

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