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On the local Cauchy problem for nonlinear hyperbolic functional differential equations

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Człapiński T.

Annales Polonici Mathematici

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1997

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tom 67

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nr 3

215-232

EN

We consider the local initial value problem for the hyperbolic partial functional differential equation of the first order (1) $Dₓz(x,y) = f(x,y,z(x,y),(Wz)(x,y),D_y z(x,y))$ on E, (2) z(x,y) = ϕ(x,y) on [-τ₀,0]×[-b,b], where E is the Haar pyramid and τ₀ ∈ ℝ₊, b = (b₁,...,bₙ) ∈ ℝⁿ₊. Using the method of bicharacteristics and the method of successive approximations for a certain functional integral system we prove, under suitable assumptions, a theorem on the local existence of weak solutions of the problem (1),(2).

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Numerical method of bicharacteristics for quasilinear hyperbolic functional differential systems

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Kropielnicka K.

Commentationes Mathematicae

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2005

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tom 45

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nr 1

EN

Classical solutions of mixed problems for first order partial functional differential systems in two independent variables are approximated in the paper with solutions of a difference problem of the Euler type. The mesh for the approximate solutions is obtained 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.

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Generalized solutions of first order partial differential functional inequalities

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Czernous W.

Commentationes Mathematicae

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2006

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tom 46

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nr 1

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.

4

On the mixed problem for quasilinear partial functional differential equations with unbounded delay

75%

Człapiński T.

Annales Polonici Mathematici

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1999

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tom 72

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nr 1

87-98

EN

We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay $D_tz(t,x) = ∑_{i=1}^n f_i(t,x,z_{(t,x)})D_{x_i}z(t,x) + h(t,x,z_{(t,x)})$, where $z_{(t,x)} ∈ X̶_0$ is defined by $z_{(t,x)}(τ,s) = z(t+τ,x+s)$, $(τ,s) ∈ (-∞,0]×[0,r]$, and the phase space $X̶_0$ satisfies suitable axioms. Using the method of bicharacteristics and the fixed-point method we prove a theorem on the local existence and uniqueness of Carathéodory solutions of the mixed problem.

Ograniczanie wyników

2 Annales Polonici Mathematici

2 Commentationes Mathematicae

2 Człapiński T.

1 Czernous W.

1 Kropielnicka K.

1 2006

1 2005

1 1999

1 1997

On the local Cauchy problem for nonlinear hyperbolic functional differential equations

Numerical method of bicharacteristics for quasilinear hyperbolic functional differential systems

Generalized solutions of first order partial differential functional inequalities

On the mixed problem for quasilinear partial functional differential equations with unbounded delay