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Delay Difference Equations and Their Applications
Jánský, Jiří ; Hilscher, Roman Šimon (oponent) ; Čermák, Libor (oponent) ; Čermák, Jan (vedoucí práce)
This thesis discusses the qualitative properties of some delay difference equations. These equations originate from the $\Theta$-method discretizations of the differential equations with a delayed argument. Our purpose is to analyse the asymptotic properties of these numerical solutions and formulate their upper bounds. We also discuss stability properties of the studied discretizations. Several illustrating examples and comparisons with the known results are presented as well.
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The Qualitative and Numerical Analysis of Nonlinear Delay Differential Equations
Dvořáková, Stanislava ; Baštinec, Jaromír (oponent) ; Šremr,, Jiří (oponent) ; Čermák, Jan (vedoucí práce)
This thesis formulates the asymptotic estimates of solutions of the so-called sublinear and superlinear differential equations with a delayed argument. These estimates are given in terms of auxiliary functional equations and inequalities. Further this thesis discusses the qualitative properties of some delay difference equations originating from discretizations of studied differential equations. We also deal with the resemblances between asymptotic behaviour of solutions of given equations in the continuous and discrete form, considering general as well as particular cases. We discuss stability properties of the $\theta$-method discretizations, too. Several examples illustrating the obtained results are included in the thesis.
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Nonlinear differential equations in the framework of the Karamata theory
Bukotin, Denys ; Opluštil, Zdeněk (oponent) ; Řehák, Pavel (vedoucí práce)
The goal of the thesis is to unify and generalize known results from literature, to study asymptotic behaviour of positive regularly varying solutions to the certain type of non-linear differential equations (known as nearly-half-linear differential equations) using available tools. This work includes description of theory of regular variation, some information on non-linear differential equations of various types, detailed derivations of results related to asymptotic behaviour of the solutions and examples of application of obtained results.
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Asymptotic Properties of Solutions of the Second-Order Discrete Emden-Fowler Equation
Korobko, Evgeniya ; Galewski, Marek (oponent) ; Růžičková, Miroslava (oponent) ; Diblík, Josef (vedoucí práce)
In the literature a differential second--order nonlinear Emden--Fowler equation $$ y'' \pm x^\alpha y^m = 0, $$ where $\alpha$ and $m$ are constants, is often investigated. This thesis deals with a discrete equivalent of the second--order Emden-Fowler differential equation $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ where $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ is an independent variable, $k_0$ is an integer and $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ is an unknown solution. In this equation, $\Delta^2u(k)=\Delta(\Delta u(k))$, $\Delta u(k)$ is the the first-order forward difference of $u(k)$, i.e., $\Delta u(k) = u(k+1)-u(k)$, and $\Delta^2 (k)$ is its second--order forward difference, i.e., $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, $\alpha$, $m$ are real numbers. The asymptotic behaviour of the solutions to this equation is discussed and the conditions are found such that there exists a power-type asymptotic: $u(k) \sim {1}/{k^s}$, where $s$ is some constant. We also discuss a discrete analogy of so-called ``blow-up'' solutions in the classical theory of differential equations, i.e., the solutions for which there exists a point $x^*$ such that $\lim_{x \to x^*} y(x) = \infty$, where $y(x)$ is a solution of the Emden-Fowler differential equation $$ y''(x) = y^s(x), $$ with $s \ne 1$ being a real number. The results obtained are compared to those already known and illustrated with examples.
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Regulární variace a její aplikace
Ženatá, Kamila ; Opluštil, Zdeněk (oponent) ; Řehák, Pavel (vedoucí práce)
Bakalářská práce se zabývá pojmem regulární variace a jejími aplikacemi v různých oblastech matematiky. Práce poskytuje přehled základních vlastností regulárně měnících se funkcí, pojmů s nimi souvisejících a konkrétní aplikace poznatků v diferenciálních rovnicích a nekonečných řadách.
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Nonlinear differential equations in the framework of the Karamata theory
Bukotin, Denys ; Opluštil, Zdeněk (oponent) ; Řehák, Pavel (vedoucí práce)
The goal of the thesis is to unify and generalize known results from literature, to study asymptotic behaviour of positive regularly varying solutions to the certain type of non-linear differential equations (known as nearly-half-linear differential equations) using available tools. This work includes description of theory of regular variation, some information on non-linear differential equations of various types, detailed derivations of results related to asymptotic behaviour of the solutions and examples of application of obtained results.
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The Qualitative and Numerical Analysis of Nonlinear Delay Differential Equations
Dvořáková, Stanislava ; Baštinec, Jaromír (oponent) ; Šremr,, Jiří (oponent) ; Čermák, Jan (vedoucí práce)
This thesis formulates the asymptotic estimates of solutions of the so-called sublinear and superlinear differential equations with a delayed argument. These estimates are given in terms of auxiliary functional equations and inequalities. Further this thesis discusses the qualitative properties of some delay difference equations originating from discretizations of studied differential equations. We also deal with the resemblances between asymptotic behaviour of solutions of given equations in the continuous and discrete form, considering general as well as particular cases. We discuss stability properties of the $\theta$-method discretizations, too. Several examples illustrating the obtained results are included in the thesis.
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Delay Difference Equations and Their Applications
Jánský, Jiří ; Hilscher, Roman Šimon (oponent) ; Čermák, Libor (oponent) ; Čermák, Jan (vedoucí práce)
This thesis discusses the qualitative properties of some delay difference equations. These equations originate from the $\Theta$-method discretizations of the differential equations with a delayed argument. Our purpose is to analyse the asymptotic properties of these numerical solutions and formulate their upper bounds. We also discuss stability properties of the studied discretizations. Several illustrating examples and comparisons with the known results are presented as well.
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