Discretized Fractional Calculus

For the numerical approximation of fractional integrals I^ f (x) = 1{ () }₀ˣ (x - s) ^{ - 1 f (s) ds (x 0) } with f (x) = x^ - 1 g (x), g smooth, we study convolution quadratures. Here approximations to I^ f (x) on the grid x = 0, h, 2h, , Nh are obtained from a discrete convolution with the values of f on the same grid. With the appropriate definitions, it is shown that such a method is convergent of order p if and only if it is stable and consistent of order p. We introduce fractional linear multistep methods: The th power of a pth order linear multistep method gives a pth order convolution quadrature for the approximation of I^. The paper closes with numerical examples and applications to Abel integral equations, to diffusion problems and to the computation of special functions.