The line method for parabolic differential equations problems in boundary layer theory and existence of periodic solutions

The (longitudinal) line method for parabolic differential equations is considered, where the space variable only is discretized. An initial-boundary value problem for a parabolic differential equation is transformed into an initial value problem for a system of ordinary differential equations of first order. Using the theory of ordinary and parabolic differential inequalities, constructive existence proofs for nonlinear problems by means of the line method can be given. Recent work on several problems in boundary layer theory along these lines is described. The main part of this exposition deals with periodic solutions of the equation ut=uxx+f(t,x,u) (periodic means periodic in t, f is assumed to be periodic). Under suitable conditions on f, there exists exactly one periodic solution u. It is shown that for the corresponding line method approximation, there exists also exactly one periodic solution. The approximations converge uniformly to u, their t-derivatives converge uniformly to ut, their first and second order differences in the x-direction converge uniformly to ux and uxx, respectively.