We present a novel approach for deriving analytical solutions to transport equations expressed in similarity variables. We apply a fixed-point iteration procedure to these transformed equations by formally solving for the highest derivative term and then integrating to obtain an expression for the solution in terms of a previous estimate. We are able to analytically obtain the Lipschitz condition for this iteration procedure and, from this (via requirements for convergence given by the contraction mapping principle), deduce a range of values for the outer limit of the solution domain, for which the fixed-point iteration is guaranteed to converge
