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Implicit difference schemes for quasilinear parabolic functional equations

Matusik M.

Demonstratio Mathematica

2012

Vol. 45, nr 4

869-886

We present a new class of numerical methods for quasilinear parabolic functional differential equations with initial boundary conditions of the Robin type. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show 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 for given functions with respect to functional variables. Results obtained in the paper can be applied to differential equations with deviated variables and to differential integral problems.

Implicit difference methods for Hamilton-Jacobi differential functional equations

Kępczyńska A.

Demonstratio Mathematica

2007

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 nonlinear parabolic functional differential systems

Kropielnicka K.

Demonstratio Mathematica

2006

Vol. 39, nr 3

711-728

Classical solutions of initial boundary value problems are approximated in the paper by solutions of implicit difference schemes. A convergence analysis for methods is presented. It is shown by an example that the new methods are considerably better than the explicit schemes. The proof of the stability is based on a theorem on difference inequalities. Nonlinear estimates of the Perron type for given functions with respect to functional variables are used. The results obtained in the paper can be applied to differential integral problems and to equations with deviated variables.