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1
Content available remote 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
Content available 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
Content available remote Difference methods for infinite systems of quasilinear parabolic functional differential equations
Jaruszewska-Walczak D.
Demonstratio Mathematica
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2010
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Vol. 43, nr 1
213-230
EN
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.
4
Content available remote 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.
5
Content available remote Generalized solutions of first order partial differential functional inequalities
Czernous W.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2006
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Vol. 46, [Z] 1
25-36
EN
The paper deals with initial boundary value problems for nonlinear first order partial differential functional equations. A theorem on the uniqueness of generalized solutions is proved. It is based on a comparison result for functional differential inequalities in the Carathéodory sense. A theorem on generalized solutions of functional differential inequalities is presented.
6
Content available remote On the structure of solution set of an integro-differential equation in Banach spaces
Roszak M.
Demonstratio Mathematica
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2005
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Vol. 38, nr 2
341--348
EN
In this paper we shall present two existence theorems for local solutions of an initial value problem for nonlinear integro-differential equation in a Banach space.
7
Content available remote 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.
8
Content available remote Initial value problems for second-order integro-differential equations on unbounded domains in Banach space
Liu Y.
Demonstratio Mathematica
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2005
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Vol. 38, nr 2
349--364
EN
In this paper, the Monch fixed point theorem is used to investigate the existence of solutions of initial value problem (IVP, for short) for second order nonlinear integro-differential equations on infinite intervals in a Banach space. At the same time, the uniqueness of solution for IVP is obtained also.
9
Content available remote 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.
10
Content available remote Differential difference inequalities generated by infinite systems of parabolic functional differential equations
Kozieł S.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2004
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Vol. 44, nr 1
99--126
EN
The paper deals with the initial boundary value problem for infinite systems of parabolic functional differential equations. A comparison theorem concerning infinite systems of differential difference inequalities generated by the original problem is proved. The comparison result is used in an existence theorem and in investigating the stability of the numerical method of lines. A theorem on the error estimate of the metod is given.
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