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Difference methods for infinite systems of quasilinear parabolic functional differential equations

Jaruszewska-Walczak D.

Demonstratio Mathematica

2010

Vol. 43, nr 1

213-230

Classical solutions of initial boundary value problems for infinite systems of quasilinear parabolic differential functional equations are considered. Two type of difference schemes are constructed. We prove that solutions of infinite difference schemes approximate solutions of our differential functional problem. In the second part of the paper we show that solutions of infinite differential functional systems can be approximated by solutions of difference systems with initial boundary conditions and the systems are finite. A complete convergence analysis for the methods is presented. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given functions.

Strong maximum principles for implicit parabolic functional-differential problems together with nonstandard inequalities with sums

Byszewski L.

Demonstratio Mathematica

2005

Vol. 38, nr 4

857--866

The aim of the paper is to give strong maximum principles for implicit parabolic functional - differential problems together with nonstandard inequalities with sums in relatively arbitrary (n + 1)-dimensional time-space sets more general than the cylindrical domain. The results obtained can be applied in the theory of diffusion and in the theory of heat conduction.