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Implicit difference methods for Hamilton-Jacobi differential functional equations

100%

Kępczyńska A.

2007

tom Vol. 40, nr 1

125-150

Classical solutions of the local Cauchy problem on the Haar pyramid are approximated in the paper by solutions of suitable quasilinear systems of difference functional equations. The numerical methods are difference schemes which are implicit with respect to time variable. A complete convergence analysis for the methods is given and it is shown that the new methods are considerable better than the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type. Numerical examples are given.

Implicit difference methods for infinite systems of hyperbolic functional differential equations

100%

Szafrańska A.

tom Vol. 50, [Z] 1

73-86

The paper deal with classical solutions of initial boundary value problems for infinite systems of nonlinear differential functional equations. Two types of difference schemes are constructed. First we show that solutions of our differential problem can be approximated by solutions of infinite difference functional schemes. In the second part of the paper we proof that solutions of finite difference systems approximate the solutions of aur differential problem. We give a complete convergence analysis for both types of difference methods. We adopt nonlinear estimates of the Perron type for given functions with respect to the functional variable. The proof of the stability is based on the comparison technique. Numerical examples are presented.