This paper describes the application of a numerical method for the solution of the nonlinear Fokker-Planck-Kolmogorov (FPK) equations of Probabilistic Elastoplasticity. We employ both a collocation and a Galerkin approach that utilize radial basis functions (RBFs) of various types, while time integration is achieved with the Crank-Nicolson scheme. The efficiency of the solution is demonstrated through simple linear elastic and elastic-perfectly plastic examples of various dimensionalities. In addition, we briefly address the accuracy and stability of the method with respect to the choice of RBF and the associated shape parameter. Finally, we provide a comparison with conventional solutions of the FPK equation to indicate the superiority of the approach.