Implicit difference methods for nonlinear first order partial functional differential systems

Abstract

Initial problems for nonlinear hyperbolic functional differential systems are considered. Classical solutions are approximated by solutions of suitable quasilinear systems of difference functional equations. The numerical methods used are difference schemes which are implicit with respect to the time variable. Theorems on convergence of difference schemes and error estimates of approximate solutions are presented. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type. Numerical examples are given.

Numerical methods for Hamilton Jacobi functional differential equations

“…The method was extended on functional differential equations in [10]. Our theorems on the convergence difference methods are generalizations of results presented in [1,6,10] (Chapter 5), [12,18].…”

“…The operators δ 0 and (δ 1 , …, δ n ) are defined by (14)- (18). The above difference problem is called an implicit generalized Euler method.…”

“…The papers [12,18] concern equations with first order partial derivatives and implicit dif ference schemes. Error estimates implying the convergence of implicit difference methods are obtained in these papers by using a method on difference inequalities or simple theorems on recurrent inequalities.…”

Differentiability with respect to initial functions of solutions to nonlinear hyperbolic functional differential systems

“…The following questions were considered: functional differential inequalities generated by initial or mixed problems and their applications [1,5,6,12], existence theory of classical or weak solutions of equations or finite systems with initial or initial boundary conditions [2][3][4]9,14,22] approximate solutions of functional differential problems [15][16][17]25]. Essential extensions of some ideas concerning generalized solution of Hamilton-Jacobi equations are given in [20,21] where viscosity solutions are considered.…”

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

“…Constructions and analysis of implicit difference schemes related to evolution functional differential equations can be found in [10,12,13,20]. Error estimates implying the convergence of implicit difference methods are obtained in these papers by using comparison techniques.…”