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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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Mathematical modeling of population problems in biology
Čampulová, Martina ; Opluštil, Zdeněk (referee) ; Čermák, Jan (advisor)
This bachelor´s thesis deals with the modeling of population problems in biology. The aim of this thesis is to mention some basic models describing dynamics of the evolution of one or two populations. Models mentioned in this thesis are described by first-order ordinary differential equations. Exploring the evolution of the population brings the main question - searching for singular points (and verifying their stability) of differential equations describing the evolution of the population. Therefore the thesis also deals with these problems.
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Stability analysis of systems of ordinary differential equations
Trejtnar, Miloš ; Opluštil, Zdeněk (referee) ; Tomášek, Petr (advisor)
This thesis deals with a stability analysis of the first order systems of ordinary differential equations. There are introduced some stability approaches in the thesis and they are discussed in the several examples. The attention is focused to the case of linear autonomous systems, where the classification of the singular points is realized. The thesis is closed by the application of the stability theory in mathematical model of electric current conduction in a primary and secondary coil of a transformer.
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Delay differential equations
Kráčmar, Jiří ; Vodstrčil, Petr (referee) ; Opluštil, Zdeněk (advisor)
Bachelor thesis focuses on the issue of differential equations with delay, which, unlike ordinary differential equations, contain in the unknown function argument the function of the so-called delay. Therefore, these are capable of a more exact description of certain real systems we want to convert into mathematic models. Practically, these are those systems where time delays, necessary for the reaction of the system to the change of status, occur. The presence of this delay, however, also complicates solution of such equations and sets further differences in comparison with ordinary equations. The crucial differences are described in this thesis. Also the principle is shown for the use of delay-differential equations in population growth models.
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Applications of ordinary differential equations with boundary conditions
Felixová, Lucie ; Opluštil, Zdeněk (referee) ; Čermák, Jan (advisor)
This bachelor's thesis is concerned with the applications of ordinary differential equations with boundary conditions. The aim of this thesis is to find the solution of straight bar stability under different boundary conditions (hinging, clamping and their combinations), of bended bars under horizontal loading and of straight bars on an elastic foundation (Winkler's foundation). Further, the thesis deals with the derivation of the equation for temperature field in a thin rod and for mathematical pendulum.
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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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System of autonomous differential equations
Benáčková, Jana ; Tomášek, Petr (referee) ; Opluštil, Zdeněk (advisor)
In his work dealing with applications, systems theory of autonomous differential equations in biology to the analysis model of coexistence of two populations. Mathematical models are described in general non-linear autonomous system of differential equations. I introduced the classification of types of singular points that are important for the following solutions to specific models. In the last part is an overview of the most famous models of the two populations (predator × prey) and specific models for the communities of invertebrate animals and mammals.
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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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