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1

Weighted difference schemes for systems of quasilinear first order partial functional differential equations

Szafrańska A.

Mathematica Applicanda

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2015

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Vol. 43, No. 2

225--251

EN

The paper deals with initial boundary value problems of the Dirichlet type for system of quasilinear functional differential equations. We investigate weighted difference methods for these problems. A complete convergence analysis of the considered difference methods is given. Nonlinear estimates of the Perron type with respect to functional variables for given functions are assumed. The proof of the stability of difference problems is based on a comparison technique. The results obtained here can be applied to differential integral problems and differential equations with deviated variables. Numerical examples are presented.

PL

Praca dotyczy zagadnień początkowo brzegowych typu Dirichlet’a dla układów quasiliniowych równań różniczkowo-funkcyjnych. Zamieszczona jest konstrukcja ważonych metod różnicowych dla wyjściowych zagadnień różniczkowych oraz przeprowadzona jest pełna analiza zbieżności. Niezbędne założenia obejmują oszacowania typu Perrona dla funkcji danych względem argumentów funkcyjnych. Dowód stabilności metody różnicowej opiera się na technice porównawczej. Teoretyczne rezultaty zobrazowane są na przykładzie całkowego równania różniczkowego oraz równań różniczkowych z odchylonym argumentem.

2

Difference functional inequalities and applications

Szafrańska A.

Opuscula Mathematica

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2014

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Vol. 34, no. 2

405--423

EN

The paper deals with the difference inequalities generated by initial boundary value problems for hyperbolic nonlinear differential functional systems. We apply this result to investigate the stability of constructed difference schemes. The proof of the convergence of the difference method is based on the comparison technique, and the result for difference functional inequalities is used. Numerical examples are presented.

3

Implicit difference methods for infinite systems of hyperbolic functional differential equations

Szafrańska A.

Commentationes Mathematicae

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2010

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Vol. 50, [Z] 1

73-86

EN

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.

4

Implicit difference methods for Hamilton-Jacobi differential functional equations

Kępczyńska A.

Demonstratio Mathematica

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2007

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Vol. 40, nr 1

125-150

EN

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.

5

Numerical approximations of difference functional equations and applications

Kamont Z.

Opuscula Mathematica

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2005

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Vol. 25, no. 1

109-130

EN

We give a theorem on the error estimate of approximate solutions for difference functional equations of the Volterra type. We apply this general result in the investigation of the stability of difference schemes generated by nonlinear first order partial differential functional equations and by parabolic problems. We show that all known results on difference methods for initial or initial boundary value problems can be obtained as particular cases of this general and simple result. We assume that the right hand sides of equations satisfy nonlinear estimates of the Perron type with respect to functional variables.

6

Numerical method of lines for first order partial differential equations with deviated variables

Czernous W.

Demonstratio Mathematica

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2005

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Vol. 38, nr 1

239--254

EN

Classical solutions of nonlinear initial boundary value problems are approximated in the paper by solutions of suitable quasilinear differential difference systems. The proof of the stability of the method of lines is based on a comparison technique with nonlinear estimates of the Perron type. Numerical examples are given.

7

Numerical method of bicharacteristics for quasilinear hyperbolic functional differential systems

Kropielnicka K.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2005

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Vol. 45, [Z] 1

91--109

EN

Classical solutions of mixed problems for first order partialfunctional differential systems in two independent variables areapproximated in the paper with solutions of a difference problemof the Euler type. The mesh for the approximate solutions isobtained by a numerical solving of equations of bicharacteristics.The convergence of explicit difference schemes is proved by means of consistency and stability arguments. It is assumed that given functions satisfy nonlinear estimates of the Perron type. Differential systems with deviated variables and differential integral systems can be obtained from a general model by specializing given operators.