In the present paper Lie theory of differential equations is expanded for finding symmetry geometric vector fields of Poisson equation. Similarity variables extracted from symmetries are applied in order to find reduced forms of the considered equation by using Erdélyi-Kober operator. Conservation laws of the space-space-fractional generalized Poisson equation with the Riemann-Liouville derivative are investigated via Noether's method. By means of the concept of non-linear self-adjointness, Noether's operators, formal Lagrangians and conserved vectors are computed. A collocation technique is also applied to give a numerical simulation of the problem.