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Continuous models in biology
Kozák, Michal ; Stará, Jana (advisor) ; Kučera, Milan (referee)
This Bachelor Thesis is devoted to study of conditions guaranteeing that the modelled biological system is stable from the point of view of surviving of species. First, we give a short survey of various concepts of ecological stability (persistence, permanence) and then we concentrate on permanence. The models we study are described in terms of semidynamical systems on metric spaces. In this framework we define permanence of a semidynamical system. Main part of the thesis are theorems giving sufficient conditions for permanence or non- permanence by adapting the method of Average Lyapunov Function. In the last chapter a model of aquatic population interacting with a polluted environment is considered and its permanence proved under certain conditions on coefficients. The aim of the theses is to present a survey of these notions. Moreover, the contri- bution of theses is the proof of non-permanence theorem whose part was known for difference equations, only. 34
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Stiff Systems Analysis
Šátek, Václav ; Dalík, Josef (referee) ; Horová, Ivana (referee) ; Kunovský, Jiří (advisor)
The solving of stiff systems is still a contemporary sophisticated problem. The basic problem is the absence of precise definition of stiff systems. A question is also how to detect the stiffness in a given system of differential equations. Implicit numerical methods are commonly used for solving stiff systems. The stability domains of these methods are relatively large but the order of them is low. The thesis deals with numerical solution of ordinary differential equations, especially numerical calculations using Taylor series methods. The source of stiffness is analyzed and the possibility how to reduce stiffness in systems of ordinary differential equations (ODEs) is introduced. The possibility of detection stiff systems using explicit Taylor series terms is analyzed. The stability domains of explicit and implicit Taylor series are presented. The solutions of stiff systems using implicit Taylor series method are presented in many examples. The multiple arithmetic must be used in many cases. The new suitable parallel algorithm based on implicit Taylor series method with recurrent calculation of Taylor series terms and Newton iteration method (ITMRN) is proposed.
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Stability and convergence of numerical computations
Sehnalová, Pavla ; Dalík, Josef (referee) ; Horová, Ivana (referee) ; Kunovský, Jiří (advisor)
Tato disertační práce se zabývá analýzou stability a konvergence klasických numerických metod pro řešení obyčejných diferenciálních rovnic. Jsou představeny klasické jednokrokové metody, jako je Eulerova metoda, Runge-Kuttovy metody a nepříliš známá, ale rychlá a přesná metoda Taylorovy řady. V práci uvažujeme zobecnění jednokrokových metod do vícekrokových metod, jako jsou Adamsovy metody, a jejich implementaci ve dvojicích prediktor-korektor. Dále uvádíme generalizaci do vícekrokových metod vyšších derivací, jako jsou např. Obreshkovovy metody. Dvojice prediktor-korektor jsou často implementovány v kombinacích modů, v práci uvažujeme tzv. módy PEC a PECE. Hlavním cílem a přínosem této práce je nová metoda čtvrtého řádu, která se skládá z dvoukrokového prediktoru a jednokrokového korektoru, jejichž formule využívají druhých derivací. V práci je diskutována Nordsieckova reprezentace, algoritmus pro výběr proměnlivého integračního kroku nebo odhad lokálních a globálních chyb. Navržený přístup je vhodně upraven pro použití proměnlivého integračního kroku s přístupe vyšších derivací. Uvádíme srovnání s klasickými metodami a provedené experimenty pro lineární a nelineární problémy.
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Semi - analytical computations and continuous systems simulation
Kopřiva, Jan ; Kubátová, Hana (referee) ; Novitzká,, Valerie (referee) ; Kunovský, Jiří (advisor)
The thesis deals with speedup and accuracy of numerical computation, especially when differential equations are solved. Algorithms, which are fulling these conditions are named semi-analytical. One posibility how to accelerate computation of differential equation is paralelization. Presented paralelization is based on transformation numerical solution into residue number system, which is extended to floating point computation. A new algorithm for modulo multiplication is also proposed. As application applications in solution of differential calculus are the main goal it is discussed numeric integration with modified Euler, Runge - Kutta and Taylor series method in residue number system. Next possibilities and extension for implemented residue number system are mentioned at the end.
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Polynomial Equations Roots
Tomšík, Filip ; Kopřiva, Jan (referee) ; Kunovský, Jiří (advisor)
Bachelor´s thesis purpose was been study solution algebraic and differential equation. We were studying Bairstow method, which is the most conducive to solution homogenous differential equation higher order. Implementation Bairstow method and her connection with Gauss elimination method. In the end we are performed tests on rate calculation and accuracy.
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Numerical Methods Accuracy vs Precision of Arithmetic
Kluknavský, František ; Šátek, Václav (referee) ; Peringer, Petr (advisor)
Thesis is dedicated to evaluation of roundoff impact on numerical integration methods accuracy and effectivity. Contains theoretical expectations taken from existing literature, implementation of chosen methods, experimental measurement of attained accuracy under different circumstances and their comparison with regard to time complexity. Library contains Runge-Kutta methods to order 7 and Adams-Bashforth methods to order 20 implemented using C++ templates which allow optional arbitrary-precision arithmetic. Small models with known analytic solution were used for experiments.
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Partial Differential Equations Parallel Solutions
Nečasová, Gabriela ; Šátek, Václav (referee) ; Kunovský, Jiří (advisor)
This thesis deals with the topic of partial differential equations parallel solutions. First, it focuses on ordinary differential equations (ODE) and their solution methods using Taylor polynomial. Another part is devoted to partial differential equations (PDE). There are several types of PDE, there are parabolic, hyperbolic and eliptic PDE. There is also explained how to use TKSL system for PDE computing. Another part focuses on solution methods of PDE, these methods are forward, backward and combined methods. There was explained, how to solve these methods in TKSL and Matlab systems. Computing accuracy and time complexity are also discussed. Another part of thesis is PDE parallel solutions. Thanks to the possibility of PDE convertion to ODE systems it is possible to represent each ODE equation by independent operation unit. These units enable parallel computing. The last chapter is devoted to implementation. Application enables generation of ODE systems for TKSL system. These ODE systems represent given hyperbolic PDE.
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Engineering Optimization
Kokrda, Lukáš ; Hrabec, Dušan (referee) ; Popela, Pavel (advisor)
The bachelor thesis deals with convex optimization and in particular, it processes a design of the optimal support of a loaded beam. For better understanding of the terms used, the bachelor thesis contains the brief introduction to the convex optimization problems, explanation of the basic therms of ordinary dierential equations and theory of elasticity. When the original model is built, then the results are obtained by computations in the MATLAB software.
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Optimization in engineering problems
Klepáč, Jaromír ; Popela, Pavel (referee) ; Mrázková, Eva (advisor)
This bachelor thesis deals with optimization in engineering problems. In particular, it focuses on a task of optimal dimensions design of continuously loaded beam with specic requirements on nal beam weight and rigidity. For the purpose of well understanding of all the terms used during solving of an illustrative task, the thesis contains a brief introduction into the optimization problems, ordinary dierential equations, and theory of elasticity as well. Results obtained through optimization in the GAMS software will be checked by analytic solution and compared with an alternative calculation in the ANSYS software.
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