Solutions obtained by expansion in a series of spatial modes and by an iterative method are compared for both space and space-time problems. In the space problem, the modal expansion is used to justify the iterative results. A useful nonlinear transformation is introduced to aid in solving multi-mode approximations. The space-dependent fast adiabatic excursion model, or Fuchs-Nordheim model, is solved by a novel iterative approach. This iterative solution is valid for large disturbances, as well as small, thus improving results obtained by approximate modal expansions. The derivation of the space-independent Fuchs-Nordheim model from the space-dependent equation is shown to follow in a more straightforward manner than derivations based on modal approximations.