Fractional Calculus in Hydrologic Modeling: A Numerical Perspective

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Fractional derivatives can be viewed either as a handy extension of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Levy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Levy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical ... continued below

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Benson, David A.; Meerschaert, Mark M. & Revielle, Jordan January 1, 2012.

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Fractional derivatives can be viewed either as a handy extension of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Levy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Levy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus.

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  • Journal Name: Advances in Water Resources; Journal Volume: in press

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  • Report No.: DOE/ER/15841-5
  • Grant Number: FG02-07ER15841
  • DOI: 10.1016/j.advwatres.2012.04.005 | External Link
  • Office of Scientific & Technical Information Report Number: 1051420
  • Archival Resource Key: ark:/67531/metadc828334

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  • January 1, 2012

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  • May 19, 2016, 9:45 a.m.

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  • June 20, 2016, 12:47 p.m.

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Benson, David A.; Meerschaert, Mark M. & Revielle, Jordan. Fractional Calculus in Hydrologic Modeling: A Numerical Perspective, article, January 1, 2012; United States. (digital.library.unt.edu/ark:/67531/metadc828334/: accessed November 22, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.