Convergence of the method of lines for parabolic differential-functional equations

Parabolic differential-functional equations with initial-boundary conditions of the Dirichlet type are studied. Spatial derivatives occurring in the original problems are replaced by suitable differences and the problem is transformed into an initial-boundary value problem for a system of ordinary differential-functional equations. The Perron type estimation for the right hand side of the original equation with respect to the functional argument is assumed.