Fractional diffusion on bounded domains

Defterli, Ozlem; D'Elia, Marta; Du, Qiang; Gunzburger, Max Donald; Lehoucq, Richard B.; Meerschaert, Mark M.

doi:10.1515/fca-2015-0023

Title: Fractional diffusion on bounded domains

Abstract

We found that the mathematically correct specification of a fractional differential equation on a bounded domain requires specification of appropriate boundary conditions, or their fractional analogue. In this paper we discuss the application of nonlocal diffusion theory to specify well-posed fractional diffusion equations on bounded domains.

Authors:

Defterli, Ozlem ^[1]; D'Elia, Marta ^[2]; Du, Qiang ^[3]; Gunzburger, Max Donald ^[4]; Lehoucq, Richard B. ^[2]; Meerschaert, Mark M. ^[5]

Michigan State Univ., East Lansing, MI (United States); Cankaya Univ., Ankara (Turkey)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Columbia Univ., New York, NY (United States); Pennsylvania State Univ., State College, PA (United States)
Florida State Univ., Tallahassee, FL (United States)
Michigan State Univ., East Lansing, MI (United States)

Publication Date:: Fri Mar 13 00:00:00 EDT 2015

Research Org.:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1183102

Report Number(s):: SAND-2014-17064J
Journal ID: ISSN 1311-0454; 537034

Grant/Contract Number:: AC04-94AL85000

Resource Type:: Journal Article: Accepted Manuscript

Journal Name:: Fractional Calculus and Applied Analysis

Additional Journal Information:: Journal Volume: 18; Journal Issue: 2; Journal ID: ISSN 1311-0454

Publisher:: de Gruyter

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; fractional diffusion; boundary value problem; nonlocal diffusion; well-posed equation

Citation Formats


                    Defterli, Ozlem, D'Elia, Marta, Du, Qiang, Gunzburger, Max Donald, Lehoucq, Richard B., and Meerschaert, Mark M. Fractional diffusion on bounded domains.  United States: N. p., 2015. 
        Web.  doi:10.1515/fca-2015-0023.

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                    Defterli, Ozlem, D'Elia, Marta, Du, Qiang, Gunzburger, Max Donald, Lehoucq, Richard B., & Meerschaert, Mark M. Fractional diffusion on bounded domains.  United States.  https://doi.org/10.1515/fca-2015-0023

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                    Defterli, Ozlem, D'Elia, Marta, Du, Qiang, Gunzburger, Max Donald, Lehoucq, Richard B., and Meerschaert, Mark M. 2015.  
        "Fractional diffusion on bounded domains".  United States.  https://doi.org/10.1515/fca-2015-0023.  https://www.osti.gov/servlets/purl/1183102.

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@article{osti_1183102,

  title        = {Fractional diffusion on bounded domains},

  author       = {Defterli, Ozlem and D'Elia, Marta and Du, Qiang and Gunzburger, Max Donald and Lehoucq, Richard B. and Meerschaert, Mark M.},

  abstractNote = {We found that the mathematically correct specification of a fractional differential equation on a bounded domain requires specification of appropriate boundary conditions, or their fractional analogue. In this paper we discuss the application of nonlocal diffusion theory to specify well-posed fractional diffusion equations on bounded domains.},

  doi          = {10.1515/fca-2015-0023},

  url          = {https://www.osti.gov/biblio/1183102},
  journal      = {Fractional Calculus and Applied Analysis},

  issn         = {1311-0454},

  number       = 2,

  volume       = 18,

  place        = {United States},

  year         = {Fri Mar 13 00:00:00 EDT 2015},

  month        = {Fri Mar 13 00:00:00 EDT 2015}

}

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Works referenced in this record:

Generalized Fick’s Law and Fractional ADE for Pollution Transport in a River: Detailed Derivation
journal, January 2006

Kim, Sangdan; Kavvas, M. Levent
Journal of Hydrologic Engineering, Vol. 11, Issue 1
https://doi.org/10.1061/(ASCE)1084-0699(2006)11:1(80)

A novel numerical method for the time variable fractional order mobile–immobile advection–dispersion model
journal, September 2013

Zhang, H.; Liu, F.; Phanikumar, Mantha S.
Computers & Mathematics with Applications, Vol. 66, Issue 5
https://doi.org/10.1016/j.camwa.2013.01.031

The fractional-order governing equation of Lévy Motion
journal, February 2000

Benson, David A.; Wheatcraft, Stephen W.; Meerschaert, Mark M.
Water Resources Research, Vol. 36, Issue 6
https://doi.org/10.1029/2000WR900032

Application of a fractional advection-dispersion equation
journal, February 2000

Benson, David A.; Wheatcraft, Stephen W.; Meerschaert, Mark M.
Water Resources Research, Vol. 36, Issue 6
https://doi.org/10.1029/2000WR900031

Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints
journal, January 2012

Du, Qiang; Gunzburger, Max; Lehoucq, R. B.
SIAM Review, Vol. 54, Issue 4
https://doi.org/10.1137/110833294

Works referencing / citing this record:

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Mathematische Nachrichten, Vol. 291, Issue 17-18
https://doi.org/10.1002/mana.201700111

Galerkin‐Legendre spectral method for the nonlinear Ginzburg‐Landau equation with the Riesz fractional derivative
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Fei, Mingfa; Huang, Chengming; Wang, Nan
Mathematical Methods in the Applied Sciences
https://doi.org/10.1002/mma.5852

Finite difference scheme for time‐space fractional diffusion equation with fractional boundary conditions
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Xie, Changping; Fang, Shaomei
Mathematical Methods in the Applied Sciences, Vol. 43, Issue 6
https://doi.org/10.1002/mma.6132

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Numerical Methods for Partial Differential Equations, Vol. 34, Issue 4
https://doi.org/10.1002/num.22254

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journal, September 2018

Li, Meng; Huang, Chengming
Numerical Methods for Partial Differential Equations, Vol. 35, Issue 1
https://doi.org/10.1002/num.22305

A second‐order finite difference method for fractional diffusion equation with Dirichlet and fractional boundary conditions
journal, January 2019

Xie, Changping; Fang, Shaomei
Numerical Methods for Partial Differential Equations, Vol. 35, Issue 4
https://doi.org/10.1002/num.22355

Determine a Space-Dependent Source Term in a Time Fractional Diffusion-Wave Equation
journal, March 2019

Yan, X. B.; Wei, T.
Acta Applicandae Mathematicae, Vol. 165, Issue 1
https://doi.org/10.1007/s10440-019-00248-2

Error Estimates of Spectral Galerkin Methods for a Linear Fractional Reaction–Diffusion Equation
journal, August 2018

Zhang, Zhongqiang
Journal of Scientific Computing, Vol. 78, Issue 2
https://doi.org/10.1007/s10915-018-0800-0

Time and Space Fractional Diffusion in Finite Systems
journal, March 2018

Raghavan, R.; Chen, C.
Transport in Porous Media, Vol. 123, Issue 1
https://doi.org/10.1007/s11242-018-1031-4

Identifying the Fractional Orders in Anomalous Diffusion Models from Real Data
journal, February 2018

Concezzi, Moreno; Spigler, Renato
Fractal and Fractional, Vol. 2, Issue 1
https://doi.org/10.3390/fractalfract2010014

Variational formulation and efficient implementation for solving the tempered fractional problems
text, January 2016

Deng, Weihua; Zhang, Zhijiang
arXiv
https://doi.org/10.48550/arxiv.1604.01860

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Title: Fractional diffusion on bounded domains

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