Georgia Institute of Technology College of Sciences

I will present results on numerical methods for fractional order operators, including the Caputo Fractional Derivative and the Fractional Laplacian. Fractional order systems have been of growing interest over the past ten years, with applications to hydrology, geophysics, physics, and engineering. Despite the large interest in fractional order systems, there are few results utilizing collocation methods. The numerical methods I will present rely heavily on reproducing kernel Hilbert spaces (RKHSs) as a means of discretizing fractional order operators. For the estimation of a function's Caputo fractional derivative we utilize a new RKHS, which can be seen as a generalization of the Fock space, called the Mittag-Leffler RKHS. For the fractional Laplacian, the Wendland radial basis functions are utilized.