Wyniki wyszukiwania

1

Generalized solutions to boundary value problems for quasilinear hyperbolic systems of partial differential-functional equations

100%

Człapiński T.

Annales Polonici Mathematici

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1992

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tom 57

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nr 2

177-191

EN

Generalized solutions to quasilinear hyperbolic systems in the second canonical form are investigated. A theorem on existence, uniqueness and continuous dependence upon the boundary data is given. The proof is based on the methods due to L. Cesari and P. Bassanini for systems which are not functional.

2

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

100%

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.

3

Difference methods for the Darboux problem for functional partial differential equations

100%

Człapiński T.

Annales Polonici Mathematici

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1999

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tom 71

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nr 2

171-193

EN

We consider the following Darboux problem: (1) $D_{xy}z(x,y) = f(x,y,z_{(x,y)},(D_xz)_{(x,y)},(D_yz)_{(x,y)})$, (2) z(x,y) = ϕ(x,y) on [-a₀,a] × [-b₀,b] \ (0,a] × (0,b], where $a₀,b₀ ∈ ℝ₊, a,b > 0. The operator $[0,a] × [0,b] ∋ (x,y) ↦ ω_{(x,y)} ∈ C([-a₀,0] × [-b₀,0],ℝ)$ defined by $ω_{(x,y)}(t,s) = ω(t+x,s+y)$ represents the functional dependence on the unknown function and its derivatives. We construct a wide class of difference methods for problem (1),(2). We prove the existence of solutions of implicit functional systems by means of a comparative method. We get two convergence theorems for implicit and explicit schemes, in the latter case with a nonlinear estimate with respect to the third variable. We give numerical examples to illustrate these results.

4

On the local Cauchy problem for nonlinear hyperbolic functional differential equations

100%

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).

5

On existence and uniqueness of solutions of nonlocal problems for hyperbolic differential-functional equations in two independent variables

100%

Człapiński T.

Annales Polonici Mathematici

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1997

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tom 67

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nr 3

205-214

EN

We seek for classical solutions to hyperbolic nonlinear partial differential-functional equations of the second order. We give two theorems on existence and uniqueness for problems with nonlocal conditions in bounded and unbounded domains.

6

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Second order evolution differential functional equations with infinite delay

64%

Człapiński T., Karpowicz A.

Commentationes Mathematicae

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2018

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tom 58

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nr 1-2

EN

We consider a second order semilinear functional evolution equation with infinite delay in a Banach space. We prove the existence of mild solutions for this equation using the measure of noncompactness technique and the Schauder fixed point theorem.

Ograniczanie wyników

5 Annales Polonici Mathematici

1 Commentationes Mathematicae

6 Człapiński T.

1 Karpowicz A.

1 2018

2 1999

2 1997

1 1992

Generalized solutions to boundary value problems for quasilinear hyperbolic systems of partial differential-functional equations

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

Difference methods for the Darboux problem for functional partial differential equations

On the local Cauchy problem for nonlinear hyperbolic functional differential equations

On existence and uniqueness of solutions of nonlocal problems for hyperbolic differential-functional equations in two independent variables

Second order evolution differential functional equations with infinite delay