Discretized Fractional Calculus

For the numerical approximation of fractional integrals $I^\alpha f(x) = \frac{1}{{\Gamma (\alpha )}}\int_0^x {(x - s)^{\alpha - 1} f(s)ds\qquad (x \geqq 0)} $ with $f(x) = x^{\beta - 1} g(x)$, g smooth, we study convolution quadratures. Here approximations to $I^\alpha f(x)$ on the grid $x = 0,h,2h, \cdots ,Nh$ 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 $\alpha $th power of a pth order linear multistep method gives a pth order convolution quadrature for the approximation of $I^\alpha $. The paper closes with numerical examples and applications to Abel integral equations, to diffusion problems and to the computation of special functions.