Classical solutions of initial boundary value problems are approximated by solutions of associated differential difference problems. A method of lines for an unknown function for the original problem and for its partial derivatives with respect to spatial variables is constructed. A complete convergence analysis for the method is given. A stability result is proved by using differential inequalities with nonlinear estimates of the Perron type for the given operators.
A discretization in time of the method of lines considered in this paper leads to new difference schemes for the original problem. It is shown by examples that the new method is considerably better than the classical schemes.