# Fractional Calculus in Hydrologic Modeling: A Numerical Perspective

### Description

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

### Creation Information

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

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## What

### Description

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.

### Source

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

### Identifier

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• Report No.: DOE/ER/15841-5
• Grant Number: FG02-07ER15841
• Office of Scientific & Technical Information Report Number: 1051420

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## When

### Creation Date

• January 1, 2012

### Added to The UNT Digital Library

• May 19, 2016, 9:45 a.m.

### Description Last Updated

• June 20, 2016, 12:47 p.m.

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