This talk is dedicated to the derivation and approximation of absorbing layers for evolution space fractional PDEs. Within this approach, widely used powerful Fourier solvers based on FFTs can be adapted without much effort to compute Initial Boundary Value Problems (IBVP) for well-posed fractional equations with absorbing boundary layers. We analyze mathematically the method and propose some illustrating numerical experiments, including fractional diffusion and Schr\"odinger

equations. This is a joint work with X. Antoine (IECL, Universite de Lorraine).