By studying a two-dimensional flow problem, the consequences of employing approximation methods in ignition problems are elucidated. Asymptotic and local similarity solutions are obtained for flows over an adiabatic plate and over a perfectly conducting plate. The governing partial differential equations are transformed into a single integral equation in Von Mises space which is amenable to solutions by the Laplace method and by the assumption of a slowly varying function. To apply the technique of local similarity to the present problem, the usual boundary layer similarity transformation is used. The results are compared with an available exact numerical solution.