Numerical Method of Lines for First Order Partial Differential-Functional Equations

We consider the Cauchy problem for a nonlinear equation on the Haar pyramid. By using a discretization with respect to spatial variables, the partial functional-differential equation is transformed into a system of ordinary functional-differential equations. We investigate the question of under what conditions the classical solutions of the original problem are approximated by solutions of associated systems of ordinary functional-differential equations. The proof of the convergence of the method of lines is based on the differentialinequalities technique. A numerical example is given. Differential equations with retarded variables and differential-integral equations are particular cases of a general model considered in the paper.