In this chapter, we present finite difference methods for solving the superdiffusive model introduced in the second chapter, which is related to Lévy flights. The problem we focus on is the Cauchy problem, defined on the real line and subject to an initial condition. We begin by introducing a first order numerical method, and then, to achieve a higher order of accuracy, we introduce another type of approximation, which is second order, for the fractional integrals and derivatives.