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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.

2

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.

3

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.

4

Numerical approximation of first order partial differential equations with deviated variables

Baranowska A., Kamont Z.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2003

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[Z] 43(1)

1-31

EN

Classical solutions of the local Cauchy problem on the Haar pyramid are approximated in the paper by solutions of suitable quasilinear systems of difference equations. The proof of the stability of the difference problem is based on a comparison technique with nonlinear estimates of the Perron type. This new approach to the numerical solving of nonlinear equations with deviated variables is generated by a quasilinearization method for initial problems. Numerical examples are given.

5

On the nonlocal boundary value problem for first order quasilinear partial functional differential equations

Człapiński T.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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1999

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[Z] 39

57-73

EN

Quasilinear partial functional differential equations of the first order with nonlocal boundary condi-tions are considered. A theorem on the local existence and uniqueness of classical solutions is given. The method of bicharacteristics is used to transform the nonlocal boundary value problem into a functional integral equation and the Banach fixed-point principle to find its solution.