期刊
ADVANCES IN WATER RESOURCES
卷 51, 期 -, 页码 479-497出版社
ELSEVIER SCI LTD
DOI: 10.1016/j.advwatres.2012.04.005
关键词
Fractional calculus; Fractional Brownian motion; Mobile/immobile; Subordination
资金
- NSF [EAR-9980489, DMS-0139943, DMS-0417972, DMS-0539176, EAR-0749035, DMS-1025486, DMS-0803360, EAR-0823965]
- USDOE Basic Energy Sciences grant [DE-FG02-07ER15841]
- NIH [R01-EB012079-01]
Fractional derivatives can be viewed either as handy extensions 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. (C) 2012 Elsevier Ltd. All rights reserved.
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