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Periodic problem for the Duffing equation
Asante, Michael Onwona ; Řehák, Pavel (referee) ; Šremr, Jiří (advisor)
Při matematickém modelování fyzikálních systémů se používají obyčejné diferenciální rovnice různých tvarů. Diferenciální rovnice popisující tyto systémy jsou často složité nelineární rovnice, avšak pomocí vhodných aproximací nelinearity lze odvodit jednoduché rovnice zvané Duffingovy rovnice, které lze analyticky studovat. V matematickém modelování mechaniky problém hledání periodických řešení těchto Duffingových rovnic úzce souvisí s existencí periodických vibrací jeho odpovídajícího nelineárního oscilátoru. V této práci je provedena analýza řešení a existence řešení v autonomních a neautonomních případech uvažované Duffingovy rovnice s podporou simulací v MATLAB.
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The Importance of Laplace Transform in Regulation Theory
Kovářová, Karolína ; Dobrovský, Ladislav (referee) ; Dosoudilová, Monika (advisor)
This bachelor's thesis deals with the significant role of Laplace transform in regulation theory. The theoretical part is dedicated to the properties of this integral transform and the description of the control system. The usage of Laplace transform in search of process variable, step and impulse function is demonstrated on specific examples of oscillation theory. The final part offers a view on Laplace transform using MATLAB software.
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Periodic solutions to nonautonmous Duffing equation
Zamir, Qazi Hamid ; Řehák, Pavel (referee) ; Šremr, Jiří (advisor)
Ordinary differential equations of various types appear in the mathematical modelling in mechanics. Differential equations obtained are usually rather complicated nonlinear equations. However, using suitable approximations of nonlinearities, one can derive simple equations that are either well known or can be studied analytically. An example of such "approximative" equation is the so-called Duffing equation. Hence, the question on the existence of a periodic solution to the Duffing equation is closely related to the existence of periodic vibrations of the corresponding nonlinear oscillator.
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Fixed point theorems in the theory of differential equations
Zelina, Michael ; Pražák, Dalibor (advisor) ; Bárta, Tomáš (referee)
This thesis is devoted to show various applications of fixed point theorems on dif- ferential equations. In the beginning we use a notion of topological degree to derive several fixed points theorems, primarily Brouwer, Schauder and Kakutani-Ky Fan the- orem. Then we apply them on a wide range of relatively simple problems from ordinary and partial differential equations (ode and pde). Finally, we take a look on a few more complex problems. First is an existence of a solution to the model of mechanical os- cillator with non-monotone dependence of both displacement and velocity. Second is a solution to so called Gause predator-prey model with a refuge. The last one is cer- tain partial differential equation with a constraint which determines maximal monotone graph. 1
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Frequency Responses
Urbánek, Radim ; Kraus, Michal (referee) ; Kunovský, Jiří (advisor)
The aim of this MSc Thesis is to create a system for automatic generation of frequency characteristics of electrical circuits. These circuits are described by differential equations. A special simulator of RLC circuit has been created and frequence response, vector diagram can be generated. This system has been mainly suggested for application in education. The process of solving differential equations is based on the Taylor method. Systems in general is the theoretical part of this project. Different definitions of systems their divission ,basic phenomenons and mathematical devices are described there. Next chapter deals with the mathematical devices for solving differential equations which makes the basis for description of phenomenons in these systems. There are also systems TKSL and TKSL/C. In the next chapter I was investigaty the analyze of vector diagrams for simple and more difficult circuits. I have found a solution for actual circuit by this technique. The last chapter is devoted to the frequency characteristics and descriptions of simulation program for generation the frequency characteristics.
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Differential equations in models of motion of dislocations
Vydrová, Jana ; Řehák, Pavel (referee) ; Šremr, Jiří (advisor)
This thesis deals with the differential equation which appears in the mathematical model by thermally activated motion of dislocations. It’s focused on screw dislocations in bodycentred cubic metals. Especially solves derivation of the pertinent differential equaiton and then explores properties of their solutions. To research these properties are used knowledges and techniques of qualitative theory of differential equations.
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Implementation of a Language Interpreter for Mathematical Calculations
Kobelka, Martin ; Šátek, Václav (referee) ; Veigend, Petr (advisor)
The main goal of this bachelor thesis is to design and implement the new programing language, which can be used for mathematical computations, implement the demonstration interpret of this language and design a graphical user interface for it. The user interface makes it easy to write the calculation, enables effective and clear visualization of calculation results and basic debugging of calculation. The properties of the resulting language are described in the thesis with the several experiments with the interpret, which implements a~subset of the language. Differences between designed solution and other platforms are also described in the thesis.
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Mathematics and implementations of physiologically based pharmacokinetic modeling
Rakhimov, Yestay ; Duintjer Tebbens, Erik Jurjen (advisor) ; Klemera, Petr (referee)
Charles University Faculty of Pharmacy in Hradec Kr'alov'e Department of Biophysics and Physical Chemistry Candidate: Yestay Rakhimov Supervisor: doc. Erik Jurjen Duintjer Tebbens, Ph.D. Title of diploma thesis: Mathematics and implementations of physiologically based phar- macokinetic modeling The thesis addresses some basic aspects of pharmacokinetic modeling, which is used to describe pharmacokinetic processes. Understanding these processes is important for example to determine optimal concentrations of drugs dosing. The thesis focuses on mathematical proofs of a number of pharmacokinetic equa- tions, which are often not given in standard books. The derived equations are illustrated with numerical experiments for a particular drug in the software PharmCalcCl and MAT- LAB. 4
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Lojasiewicz inequality for various classes of functions
Surma, Martin ; Bárta, Tomáš (advisor) ; Zelený, Miroslav (referee)
Bachelor thesis pursue the Łojasiewicz inequality. The Łojasiewicz inequality is proved here for generalized Morse-Bott functions and for functions with simple normal crossings. Further on, we study optimality of the Łojasiewicz exponent for those functions. In the last chapter, there are possible applications of the Łojasiewicz inequality to certain gradient-like differential equation stated and proved, such as the theorem on convergence of its solution. There is also shown how one can use the Łojasiewicz exponent to estimate the rate of the convergence. 1
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