Explicit and Implicit Difference Methods for Quasilinear First Order Partial Functional Differential Equations

Initial boundary value problems of the Dirichlet type for quasilinear functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both classes of the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. Error estimates for both methods are presented. Interpolating operators corresponding to functional variables are constructed.