Fractional calculus in hydrologic modeling: A numerical perspective

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 Lévy 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 Lévy) 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.

► The link is defined between the most general forms of Levy motion and fractional diffusion equations. ► Fractional Brownian motion is generalized in multiple dimensions based on inverse fractional derivative operators. ► Lagrangian and Eulerian numerical procedures for solving fractional advection-dispersion equations are reviewed. ► Procedures for generating and conditioning multi-dimensional operator-scaling fractional Brownian motion are developed. ► Applications of the fractional calculus to problems in hydrology are reviewed.