Vector Grünwald Formula for Fractional Derivatives

Fractional derivatives have been around for centuries [22, 26] but recently they have found new applications in physics [2, 6, 7, 9, 15, 18, 19, 29], hydrology [1, 4, 5, 10, 14, 28], and finance [24, 25, 27]. Analytical solutions of ordinary fractional differential equations [22, 23] and partial fractional differential equations [8, 16] are now available in some special cases. But the solution to many fractional differential equations will have to rely on numerical methods, just like their integer-order counterparts. Numerical solutions of fractional differential equations require a numerical estimate of the fractional derivative. In one dimension, this estimate is called the Grünwald formula [20, 22, 26]. A variant of this formula has been used to develop practical numerical methods for solving certain fractional partial differential equations that model flow in porous media [21, 30]. The purpose of this paper is to develop a multivariable analogue of the Grünwald formula for estimating multidimensional fractional derivatives, so that the results in [21, 30] can be extended to two and three dimensional flow regimes. The scalar Grünwald formula is a discrete approximation to the fractional derivative by a fractional difference quotient