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1

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

Classical solutions of mixed problems for quasilinear first order PFDEs on a cylindrical domain

Czernous W.

Opuscula Mathematica

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2014

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Vol. 34, no. 2

291--310

EN

We abandon the setting of the domain as a Cartesian product of real intervals, customary for first order PFDEs (partial functional differential equations) with initial boundary conditions. We give a new set of conditions on the possibly unbounded domain Ω with Lipschitz differentiable boundary. Well-posedness is then reliant on a variant of the normal vector condition. There is a neighbourhood of ∂Ω with the property that if a characteristic trajectory has a point therein, then its every earlier point lies there as well. With local assumptions on coefficients and on the free term, we prove existence and Lipschitz dependence on data of classical solutions on (0,c)×Ω to the initial boundary value problem, for small c. Regularity of solutions matches this domain, and the proof uses the Banach fixed-point theorem. Our general model of functional dependence covers problems with deviating arguments and integro-differential equations.

3

Pseudospectral method for semilinear partial functional differential equations

Czernous W.

Opuscula Mathematica

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2010

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Vol. 30, no. 2

133-145

EN

We present a convergence result for two spectral methods applied to an initial boundary value problem with functional dependence of Volterra type. Explicit condition of Courant-Friedrichs-Levy type is assumed on time step τ and the number N of collocation points. Stability statements and error estimates are written using continuous norms in weighted Jacobi spaces.

4

Classical solutions of initial problems for quasilinear partial functional differential equations of the first order

Czernous W.

Opuscula Mathematica

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2006

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Vol. 26, no. 1

13-29

EN

We consider the initial problem for a quasilinear partial functional differential equation of the first order [formula], z(t, x) = varphi(t, x) ((t, x) ∈ [-h0, 0] x Rn) where z(t, x) : [-h0, 0] x [-h, h] → R is a function defined by z(t, x) (τ, ξ) = z(t + τ, + ξ) for (τ, ξ) ∈ [-h0, 0] x [-h, h]. Using the method of bicharacteristics and the fixed-point theorem we prove, under suitable assumptions, a theorem on the local existence and uniqueness of classical solutions of the problem and its continuous dependence on the initial condition.

5

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

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.