Method of lines for nonlinear first order partial functional differential equations

Classical solutions of initial problems for nonlinear functional differential equations of Hamilton Jacobi type are approximated by solutions of associated differential difference systems. A method of quasilinearization is adopted. Sufficient conditions for the convergence of the method of lines and error estimates for approximate solutions are given. Nonlinear estimates of the Perron type with respect to functional variables for given operators are assumed. The proof of the stability of differential difference problems is based on a comparison technique. The results obtained here can be applied to differential integral problems and differential equations with deviated variables.