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Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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A note on a Herzog’s and Lemmert’s paper

100%
Leszczyński H.
Commentationes Mathematicae
|
2005
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tom 45
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nr 2
EN
We give an alternative view of the results published in the Herzog’s and Lemmert’s paper "On maximal and minimal solutions for \(x'(t) = F(t, x(t), x(h(t)))\), \(x(0) = x_0\)", Comment. Math. XL (2000), 93-102. One can observe that these results can be obtained by classical (elementary) methods, instead of Tarski’s fixed point theorems in partially ordered spaces.
2
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Convergence results for unbounded solutions of first order non-linear differential-functional equations

100%
Leszczyński H.
Annales Polonici Mathematici
|
1996
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tom 64
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nr 1
1-16
EN
We consider the Cauchy problem in an unbounded region for equations of the type either $D_{t}z(t,x) = f(t,x,z(t,x),z_{(t,x)},D_{x}z(t,x))$ or $D_{t}z(t,x)= f(t,x,z(t,x),z,D_{x}z(t,x))$. We prove convergence of their difference analogues by means of recurrence inequalities in some wide classes of unbounded functions.
3
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General finite difference approximation to the Cauchy problem for nonlinear parabolic differential-functional equations

75%
Leszczyński H.
Annales Polonici Mathematici
|
1991
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tom 53
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nr 1
15-28
4
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Existence of solutions to generalized von Foerster equations with functional dependence

64%
Leszczyński H., Zwierkowski P.
Annales Polonici Mathematici
|
2004
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tom 83
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nr 3
201-210
EN
We prove the existence of solutions to a differential-functional system which describes a wide class of multi-component populations dependent on their past time and state densities and on their total size. Using two different types of the Hale operator, we incorporate in this model classical von Foerster-type equations as well as delays (past time dependence) and integrals (e.g. influence of a group of species).
5
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Existence of solutions with exponential growth for nonlinear differential-functional parabolic equations

64%
Bartłomiejczyk A., Leszczyński H.
Annales Polonici Mathematici
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2014
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tom 111
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nr 3
309-326
EN
We consider the Cauchy problem for nonlinear parabolic equations with functional dependence. We prove Schauder-type existence results for unbounded solutions. We also prove existence of maximal solutions for a wide class of differential functional equations.
6
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Stability of finite difference schemes for certain problems in biology

64%
Leszczyński H., Zwierkowski P.
Applicationes Mathematicae
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2004
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tom 31
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nr 1
13-30
EN
We consider a generalized 1-D von Foerster equation. We present two discretization methods for the initial value problem and study stability of finite difference schemes on regular meshes.
7
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On the Cauchy problem for hyperbolic functional-differential equations

64%
Karpowicz A., Leszczyński H.
Annales Polonici Mathematici
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2015
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tom 115
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nr 1
53-74
EN
We consider the Cauchy problem for a nonlocal wave equation in one dimension. We study the existence of solutions by means of bicharacteristics. The existence and uniqueness is obtained in $W^{1,∞}_{loc}$ topology. The existence theorem is proved in a subset generated by certain continuity conditions for the derivatives.
8
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Thermal ablation modeling via the bioheat equation and its numerical treatment

51%
Bartłomiejczyk A., Leszczyński H., Poliński A.
Applicationes Mathematicae
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2015
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tom 42
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nr 1
1-12
EN
The phenomenon of thermal ablation is described by Pennes' bioheat equation. This model is based on Newton's law of cooling. Many approximate methods have been considered because of the importance of this issue. We propose an implicit numerical scheme which has better stability properties than other approaches.
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