The multigroup neutron diffusion equation is solved numerically by the meshless radial basis function collocation and mesh-based finite and boundary element methods. For the collocation method, multiquadrics, inverse multiquadrics and Gaussian basis functions are utilized, whereas linear shape functions are the choice for finite and boundary element methods. External and fission source problems are studied. In the context of external source case, constant, trigonometric, and linear sources are considered. The collocation method converges exponentially which is faster than the algebraic rates of finite and boundary element methods for both problems, and it was found that by adjusting the value of the shape parameter, very high accuracies can be achieved even with large fill distances. In the fission source case, multiquadrics is found to be superior to finite and boundary elements for the determination of multiplication factor, while boundary elements gave the best result for group fluxes. A comparison of CPU times shows that, finite element method has outperformed radial basis function collocation and boundary elements. When the stability is considered, finite and boundary element methods have the advantage of being more stable than the collocation technique. (C) 2014 Elsevier Ltd. All rights reserved.