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 N0, for the unique solution of the following time-stepping scheme C∇αun = Aun + fn, n ≥ 2; (∗) u0 = u0; u1 = u1; as well as its convergence with rates to the solution of the abstract fractional Cauchy problem (∗) ∂tαu(t) = Au(t) + f(t), t > 0; u(0) = u0; u0(0) = u1; in the superdiffusive case 1 < α < 2. Here, C∇αun is the Caputo-like fractional difference operator of order α. © 2023 American Institute of Mathematical Sciences. All rights reserved.