The method of lines for parabolic differential-functional equations

Parabolic differential-functional equations with initial-boundary conditions of Dirichlet type are studied. Spatial derivatives occurring in the original problem 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. The convergence of the numerical method of lines is proved. The differential inequalities technique is applied.