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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

A finite difference method for quasi-linear and nonlinear differential functional parabolic equations with Neumann's condition

Sapa S.

Commentationes Mathematicae

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2009

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Vol. 49, [Z] 1

83-106

EN

Classical solutions of nonlinear second-order partial dierential functional equations of parabolic type with Neumann's condition are approximated in the paper by solutions of associated explicit dierence functional equations. The functional dependence is of the Volterra type. Nonlinear estimates of the generalized Perron type for given functions are assumed. The convergence and stability results are proved with the use of the comparison technique. These theorems in particular cover quasi-linear equations, but such equations are also treated separately. The known results on similar dierence methods can be obtained as particular cases of our simple result.