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1
Content available Difference problems generated by infinite systems of nonlinear parabolic functional differential equations with the Robin conditions
Czernous W., Jaruszewska-Walczak D.
Opuscula Mathematica
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2014
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Vol. 34, no. 2
311--326
EN
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.
2
Content available remote Implicit difference schemes for quasilinear parabolic functional equations
Matusik M.
Demonstratio Mathematica
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2012
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Vol. 45, nr 4
869-886
EN
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.
3
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.
4
Content available Uniqueness of solutions of a generalized Cauchy problem for a system of first order partial functional differential equations
Netka M.
Opuscula Mathematica
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2009
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Vol. 29, no. 1
69-79
EN
The paper is concerned with weak solutions of a generalized Cauchy problem for a nonlinear system of first order differential functional equations. A theorem on the uniqueness of a solution is proved. Nonlinear estimates of the Perron type are assumed. A method of integral functional inequalities is used.
5
Content available remote 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.
6
Content available remote Numerical approximations of parabolic functional differential equations on unbounded domains
Baranowska A.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2007
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Vol. 47, [Z] 2
193-205
EN
The paper is concerned with initial problems for nonlinear parabolic functional differential equations. A general class of difference methods is constructed. A theorem on the error estimate of approximate solutions for difference functional equations of the Volterra type with an unknown function of several variables is presented. The convergence of explicit difference schemes is proved by means of consistency and stability arguments. It is assumed that given function satisfy nonlinear estimates of the Perron type with respect to a functional variable. Results obtained in the paper can be applied to differential integral problems and equations with retarded variables. Numerical examples are presented.
7
Content available 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.
8
Content available remote Difference methods for infinite systems of hyperbolic functional differential equations on the Haar pyramid
Jaruszewska-Walczak D.
Opuscula Mathematica
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2004
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Vol. 24, no. 1
85-96
EN
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.
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