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Solving of partial differential equations by Fourier method
Barvenčík, Oldřich ; Opluštil, Zdeněk (referee) ; Nechvátal, Luděk (advisor)
Bachelor thesis is a survey text which deals with solving partial deferential equations by Fourier method, i.e. method when we look for a solution of (initial) boundary value problem in form of the infinite Fourier series. The key step is a hypothesis that the solution can be expressed in form with separated variables, therefore the method is sometimes called separation of variables method. The essence can be demonstrated on parabolic and hyperbolic homogeneous problems. In the thesis both types in one (space) dimension are systematically analyzed, at first homogeneous problem, then homogeneous one with non-homogeneous boundary conditions and finally completely non-homogeneous problem.
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Macroscopic traffic flow modelling
Pidrová, Kateřina ; Opluštil, Zdeněk (referee) ; Kisela, Tomáš (advisor)
This bachelor thesis is focused on macroscopic traffic flow modelling. First, we present a short introduction into the topic and basic model classification. Then we derive the continuity equation for macroscopic models and outline possible constitutive relations for a flux which determine the key features of the model. The main part of this work is focused on LWR model and its solution by the method of characteristic curves, with emphasize on shockwave creation. The thesis is concluded by an example and experiment comparing of the LWR model with real traffic on highway.
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Mathematical models of linear and nonlinear oscillators
Lovas, David ; Opluštil, Zdeněk (referee) ; Čermák, Jan (advisor)
This bachelor thesis deals with mathematical models of linear mechanical oscillators, which represent one of basic applications of ordinary differential equations. There are explained harmonic oscillators, damped oscillators and driven oscillators. The thesis discusses also superposition of oscillators and coupling of oscillators, including their synchronization.
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Asymptotic stability of systems of linear ordinary differential equations in engineering
Mašek, Jakub ; Opluštil, Zdeněk (referee) ; Tomášek, Petr (advisor)
This bachelors thesis is dealing with stability of system of linear ordinary dierential equations and specially lyapunov stability and asymtotic stability.The are established necessary concepts from the theory of stability and form systems of dierential equations at rst. Furthermore there are listed basic methods for determining the stability of linear dierential equations with constant coecients and they are compared. The next part of thesis is dedicated to trajectory in plane with focus on isolated singular points. At the end are two technical applications and they are linked sections and oscillators.
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Autonomous differential equations
Bokišová, Lenka ; Vodstrčil, Petr (referee) ; Opluštil, Zdeněk (advisor)
This bachelor's thesis is concerned with solution of autonomous dierential equations. Attention is devoted to the basic mathematical models of population growth of single species. It is here mentioned Malthus model, model with intraspecic competition and analyzed the model of population growth under predation. The acquired knowledge is applied to specic mathematical models of sheries. Here are distinguish cases where shing is a constant and depends on the size of the population. Moreover, it is studied the model of shing of sardines with special growth function. In each model is dealt with the question of stability of stationary solutions.
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Application of boundary value problems for ordinary differential equations in engineering
Zapoměl, Jakub ; Šremr, Jiří (referee) ; Opluštil, Zdeněk (advisor)
This bachelor thesis deals with the determination of the shape of the deflection line for boundary value problems in strength of materials. There are several methods for solving boundary value problems. This thesis focuses on the Green's function method. It provides a basic review of the properties of ordinary differential equations, an introduction to the Green's function method and the actual application of the findings to beam bending models. The concrete models are solved using an interactive program developed in Matlab software.
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Stability analysis of delay differential equations
Pustějovský, Michal ; Opluštil, Zdeněk (referee) ; Tomášek, Petr (advisor)
This thesis deals with asymptotic stability analysis of delayed differential equations. First we focus on introduction of this type of equations. Next we study stability of linear autonomous equations. Here we get some simple criteria of stability. The main part of the thesis is application of these criteria to a engineering problem - the model of turning tool regenerative effect. In mathematical sense, it is a initial value problem of linear delayed differential equation. Practical outcome of this thesis is a computer application written in Maple environment displaying stability region.
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Nonlinear differential equations in the framework of the Karamata theory
Bukotin, Denys ; Opluštil, Zdeněk (referee) ; Řehák, Pavel (advisor)
Cílem této diplomové práce je sjednotit a zobecnit známé výsledky z literatury, studovat asymptotické chování kladných regulárně se měnících řešení jisté třídy nelineárních diferenciálních rovnic (tzv. skoro pololineárních diferenciálních rovnic) pomocí dostupných nástrojů. Tato práce zahrnuje popis teorie regulární variace, některé informace o nelineárních diferenciálních rovnicích různých typů, detailní odvození výsledků týkajících se asymptotického chování řešení a příklady aplikace získaných výsledků.
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