Solving nonlinear parabolic problems with result verification. Part I: one-space dimensional case

We propose some numerical methods for the automatic proof of existence of weak solutions for parabolic initial boundary value problems with one space dimension. It also means that one can obtain a posteriori error bounds for the approximate solutions of the problems. Based upon Schauder's fixed-point theorem, a verification condition is formulated and, by the use of finite-element approximation and its error estimates for a simple parabolic problem, we present a numerical verification algorithm of exact solutions in a computer. Some numerical examples which are verified by the method are illustrated.