Fractional Calculus in Hydrologic Modeling: A Numerical Perspective Page: 1 of 44
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1 Fractional Calculus in Hydrologic Modeling:
2 A Numerical Perspective
3 David A. Benson
4 Hydrological Science and Engineering, Colorado School of Mines, Golden, CO, 80401, USA.
s (dbenson@mines. edu)
6 Mark M. Meerschaert
7 Department of Statistics and Probability, Michigan State University, East Lansing, MI,
s USA. (mcubed@stt.msu.edu)
9 Jordan Revielle
io Hydrological Science and Engineering, Colorado School of Mines, Golden, CO, 80401, USA.
ii (Now at Ward Petroleum Corporation, Fort Collins, CO 80521) (jkrevielle@gmail.com)
12 Abstract
13 Fractional derivatives can be viewed either as a handy extension of classical
14 calculus or, more deeply, as mathematical operators defined by natural phe-
15 nomena. This follows the view that the diffusion equation is defined as the
16 governing equation of a Brownian motion. In this paper, we emphasize that
17 fractional derivatives come from the governing equations of stable L6vy motion,
1s and that fractional integration is the corresponding inverse operator. Fractional
19 integration, and its multi-dimensional extensions derived in this way, are inti-
20 mately tied to fractional Brownian (and L6vy) motions and noises. By following
21 these general principles, we discuss the Eulerian and Lagrangian numerical so-
22 lutions to fractional partial differential equations, and Eulerian methods for
23 stochastic integrals. These numerical approximations illuminate the essential
24 nature of the fractional calculus.
25 Keywords: Fractional Calculus, fractional Brownian motion,
26 Mobile/Immobile, Subordination
27 PA CS: 02.50.Ey, 02.50.Ga, 02.70.Ns, 05.10.Gg
28 1. Introduction
29 The term "fractional calculus" refers to the generalization of integer-order
30 derivatives and integrals to rational order. This topic was first broached by
31 L'Hopital and Leibniz after the latter's co-invention of calculus in the 1700s
32 (see the excellent history by [1]). In fact, the operators can be extended toPreprint submitted to Elsevier
April 16, 2012
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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. (https://digital.library.unt.edu/ark:/67531/metadc828334/m1/1/: accessed April 26, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.