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Difference methods for infinite systems of hyperbolic functional differential equations on the Haar pyramid

100%

Jaruszewska-Walczak D.

Opuscula Mathematica

2004

tom Vol. 24, no. 1

85-96

We consider the Cauchy problem for infinite system of differential functional equations ∂tzk(t, x) = fk(t, x, z, ∂xzk(t, x)), k mem N. In the paper we consider a general class of difference methods for this problem. We prove the convergence of methods under the assumptions that given functions satisfy the nonlinear estimates of the Perron type with respect to functional variables. The proof is based on functional difference inequalities. We constructed the Euler method as an example of difference method.

On the Chaplygin method for the Darboux problem

100%

Jaruszewska-Walczak D.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2007

tom Vol. 47, [Z] 1

11-21

In the paper we deal with the Darboux problem for hyperbolic functional differntial equations. We give the sufficient conditions for the existence of the sequence {z^(m)} such that if z is a classical solution of the original problem then {z^(m)} is uniformly convergent to z. The convergence that we get is of the Newton type.

One-step difference methods for mixed type differential-functional equations

100%

Jaruszewska-Walczak D.

Demonstratio Mathematica

1999

tom Vol. 32, nr 1

105-116

Difference methods for infinite systems of quasilinear parabolic functional differential equations

100%

Jaruszewska-Walczak D.

Demonstratio Mathematica

2010

tom 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.

Difference problems generated by infinite systems of nonlinear parabolic functional differential equations with the Robin conditions

63%

Czernous W. , Jaruszewska-Walczak D.

Opuscula Mathematica

2014

tom Vol. 34, no. 2

311--326

We consider the classical solutions of mixed problems for infinite, countable systems of parabolic functional differential equations. Difference methods of two types are constructed and convergence theorems are proved. In the first type, we approximate the exact solutions by solutions of infinite difference systems. Methods of second type are truncation of the infinite difference system, so that the resulting difference problem is finite and practically solvable. The proof of stability is based on a comparison technique with nonlinear estimates of the Perron type for the given functions. The comparison system is infinite. Parabolic problems with deviated variables and integro-differential problems can be obtained from the general model by specifying the given operators.