Time discretization and convergence to superdiffusion equations via poisson distribution

Let A be a closed linear operator defined on a complex Banach space X. We show a novel representation, using strongly continuous families of bounded operators defined on No, for the unique solution of the following time-stepping scheme (*) {C(del alpha)u(n) = Au-n + f(n), n >= 2, u(0) = u(0); u(1) = u(1); as well as its convergence with rates to the solution of the abstract fractional Cauchy problem (*) {partial derivative(alpha)(t)u(t) = Au(t) + f(t), t > 0; u(0) = u(0); u'(0) = u(1); in the superdiffusive case 1 < alpha < 2. Here, c del(alpha)u(n) Vaun is the Caputo-like fractional difference operator of order alpha.