Finite difference approximations for fractional advection-dispersion flow equations

doi:10.1016/J.CAM.2004.01.033

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Finite difference approximations for fractional advection-dispersion flow equations

Journal Article•DOI•

Finite difference approximations for fractional advection-dispersion flow equations

Mark M. Meerschaert¹, Charles Tadjeran¹•Institutions (1)

01 Nov 2004-Journal of Computational and Applied Mathematics (Elsevier Science Publishers B. V.)-Vol. 172, Iss: 1, pp 65-77

TL;DR: In this paper, the authors developed practical numerical methods to solve one dimensional fractional advection-dispersion equations with variable coefficients on a finite domain and demonstrated the practical application of these results is illustrated by modeling a radial flow problem.

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About: This article is published in Journal of Computational and Applied Mathematics.The article was published on 2004-11-01 and is currently open access. It has received 1334 citations till now. The article focuses on the topics: Fractional calculus & Finite difference.

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Journal Article•DOI•

Finite difference/spectral approximations for the time-fractional diffusion equation

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Yumin Lin¹, Chuanju Xu¹•Institutions (1)

Xiamen University¹

01 Aug 2007-Journal of Computational Physics

TL;DR: It is proved that the full discretization is unconditionally stable, and the numerical solution converges to the exact one with order O(@Dt^2^-^@a+N^- ^m), where @Dt,N and m are the time step size, polynomial degree, and regularity of the exact solution respectively.

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1,420 citations

Journal Article•DOI•

Finite difference approximations for two-sided space-fractional partial differential equations

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Mark M. Meerschaert¹, Charles Tadjeran²•Institutions (2)

University of Otago¹, University of Nevada, Reno²

01 Jan 2006-Applied Numerical Mathematics

TL;DR: In this paper, the authors examined some practical numerical methods to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain, and the stability, consistency, and (therefore) convergence of the methods are discussed.

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836 citations

Additional excerpts

...…i = 1, . . . , K − 1 and j = 1, . . . , K − 1 are defined by Ai,j = 1 − (ξi + ηi)g1 for j = i −(ξig2 + ηig0) for j = i − 1 −(ξig0 + ηig2) for j = i + 1 −ξigi−j+1 for j < i − 1 −ηigj−i+1 for j > i + 1 while A0,0 = 1, A0,j = 0 for j = 1, . . . , K, AK,K = 1, and AK,j = 0 for j = 0, . . . , K − 1....
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Journal Article•DOI•

A second-order accurate numerical approximation for the fractional diffusion equation

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Charles Tadjeran¹, Mark M. Meerschaert², Hans-Peter Scheffler¹•Institutions (2)

University of Nevada, Reno¹, University of Otago²

20 Mar 2006-Journal of Computational Physics

TL;DR: It is shown that the fractional Crank-Nicholson method based on the shifted Grunwald formula is unconditionally stable and compared with the exact analytical solution for its order of convergence.

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557 citations

Cites background or methods from "Finite difference approximations fo..."

...The standard Grünwald estimates generally yield unstable finite difference equations regardless of whether the resulting finite difference method is an explicit or an implicit system, see [9] for related discussion....
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...Several different first-order accurate numerical methods to solve fractional diffusion equations have been presented before [3,4,6,7,9,10]....
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Journal Article•DOI•

A class of second order difference approximations for solving space fractional diffusion equations

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Wenyi Tian¹, Han Zhou¹, Weihua Deng², Weihua Deng¹•Institutions (2)

Lanzhou University¹, Hong Kong Baptist University²

06 Jan 2015-Mathematics of Computation

TL;DR: A class of second order approximations, called the weighted and shifted Grunwald difference (WSGD) operators, are proposed for Riemann-Liouville fractional derivatives, with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions.

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Abstract: A class of second order approximations, called the weighted and shifted Grunwald difference (WSGD) operators, are proposed for Riemann-Liouville fractional derivatives, with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions. The stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established. Several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order, and the numerical results for variable coefficients problem are also presented.

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546 citations

Cites methods from "Finite difference approximations fo..."

...The Riemann-Liouville fractional derivative can be discretized by the standard Grünwald-Letnikov form ula [18] with only the first order accuracy, but the difference scheme based on the Grünwald-Letnikov form ula for time dependent problems is unstable [12]....
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...In [12], the shifted Grünwald difference operator Ah,pu(x) = 1 hα ∞ ∑...
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Journal Article•DOI•

Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation

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Fawang Liu¹, Fawang Liu², Pinghui Zhuang¹, Vo Anh², Ian Turner², Kevin Burrage³ - Show less +2 more•Institutions (3)

Xiamen University¹, Queensland University of Technology², University of Queensland³

01 Aug 2007-Applied Mathematics and Computation

TL;DR: Using mathematical induction, it is proved that the IDM is unconditionally stable and convergent, but the EDM is conditionally stable and Convergent.

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514 citations

Cites background or methods from "Finite difference approximations fo..."

...For the space fractional derivatives Dx uðxi; tkþ1Þ and Dxuðxi; tkþ1Þ, we adopted the Grünwald formula [11] and the shift Grünwald formula at level tk+1 [7], respectively Dx uðxi; tkþ1Þ 1⁄4 1 h Xi j1⁄40 g j uðxi jh; tkþ1Þ þOðhÞ; ð9Þ Dxuðxi; tkþ1Þ 1⁄4 1 h Xiþ1 j1⁄40 g j uðxi ðj 1Þh; tkþ1Þ þOðhÞ; ð10Þ...
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...[7] presented practical numerical methods to solve the one-dimensional space fractional ADE with variable coefficients on a finite domain....
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...The following lemma has been proved previously [7,14]: Lemma 1....
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References

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Kenneth S. Miller, Bertram Ross

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TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.

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Abstract: Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.

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TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.

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Abstract: Fractional integrals and derivatives on an interval fractional integrals and derivatives on the real axis and half-axis further properties of fractional integrals and derivatives other forms of fractional integrals and derivatives fractional integrodifferentiation of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations fo the first kind with special function kernels applications to differential equations.

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7,096 citations

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Difference Methods for Initial-Value Problems.

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Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

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Jean-Philippe Bouchaud¹, Antoine Georges¹•Institutions (1)

Centre national de la recherche scientifique¹

01 Nov 1990-Physics Reports

TL;DR: In this article, the authors consider the specific effects of a bias on anomalous diffusion, and discuss the generalizations of Einstein's relation in the presence of disorder, and illustrate the theoretical models by describing many physical situations where anomalous (non-Brownian) diffusion laws have been observed or could be observed.

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3,383 citations