A continuous variant for Grünwald–Letnikov fractional derivatives

The names of Grünwald and Letnikov are associated with discrete convolutions of mesh h, multiplied by h−α. When h tends to zero, the result tends to a Marchaud’s derivative (of the order of α) of the function to which the convolution is applied. The weights of such discrete convolutions form well-defined sequences, proportional to k−α−1 near infinity, and all moments of integer order r<α are equal to zero, provided α is not an integer. We present a continuous variant of Grünwald–Letnikov formulas, with integrals instead of series. It involves a convolution kernel which mimics the above-mentioned features of Grünwald–Letnikov weights. A first application consists in computing the flux of particles spreading according to random walks with heavy-tailed jump distributions, possibly involving boundary conditions.