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Spline-base functions for the soulution of boundary-value problems
Horčička, Martin ; Dolejší, Vít (advisor) ; Feistauer, Miloslav (referee)
Solving the Poisson equation using finite element method with a basis com- posed of natural cubic splines. In this thesis we introduce the notion of weak derivatives, Sobolev spaces and formulate the weak form of the Poisson equation in order to build up to the finite element method. Furthermore, the thesis contains a construction of a natural cubic spline and a description of the used basis. The computed solution approximates well the exact solution, especially if the right side satisfies certain conditions. 1
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Mathematical Methods in Economics
Válka, Vojtěch ; Doubravský, Karel (referee) ; Novotná, Veronika (advisor)
This thesis is focused on problems of ordinary differential equations of the first degree. The first part is dedicated to theory of differential equations. In the second part, solved and unsolved examples of individual types of differential equations are presented. In closing, there are few examples of economic applications. The thesis serve as a study material for students of economic faculty.
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Mathematical analysis of bone and vascular remodelling model
Matajová, Adéla ; Maršík, František (advisor) ; Souček, Ondřej (referee)
Bone is a tissue that is constantly being renewed during the whole life. This complex biological process, controling among others adaptation to environ- mental loads, is called bone remodelling. It is due to this complexity that the process hasn't been fully biomechanically described yet. However, sev- eral mathematical models of bone remodelling have been conjectured, one of which we will introduce and analyze in this thesis. The model describes bone metabolism by five chemical equations. Using the biothermodynamical laws we will derive from these equations a system of ordinary differential equations. Then we will effectuate a qualitative analysis, while focusing on existence, uniqueness and stability of a stationary solution. Finally the im- pact of the mechanical loading on bone remodelling will be outlined.We will also mention the relation with vascular remodelling. 1
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Filippov dynamical systems with applications
Šimonová, Dorota ; Janovský, Vladimír (advisor) ; Ratschan, Stefan (referee)
The thesis is motivated by problems of contact mechanics with friction. At the beginning we describe a class of piecewise smooth systems with discontinuous vector field called Filippov systems. We also show how to solve them. The rest of this thesis is focused on applications, especially dry friction model and finite element model of Coulomb friction with one contact point. We propose a technique for simulation of the second mentioned model which combines sovling methods for Filippov systems and impact oscillators. Powered by TCPDF (www.tcpdf.org)
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Transformations of ODEs into gradient systems in stationary points
Bílý, Michael ; Bárta, Tomáš (advisor) ; Spurný, Jiří (referee)
This bachelor thesis follows article by Bárta, Chill a Fašangová [1]. It is proven there that every ordinary differential equation with a strict Lyapunov function is in fact a gradient system for certain Riemannian metric on the set of all non-equilibrium points. We will try to determine necessary and sufficient conditions for this Riemannian metric to have continuous extension to isolated equilibrium point so that the ODE is gradient system on the whole domain. Powered by TCPDF (www.tcpdf.org)
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On a model of corruption in a democratic society
Splítek, Martin ; Janovský, Vladimír (advisor) ; Mlčoch, Lubomír (referee)
The aim of this work is to study the behavior of serious social pheno- menon - corruption, and we do this through a mathematical model of corruption in a democratic society, published in [1]. The model is a dynamical system of three differential equations, specified by three variables and ten parameters. The model is studied by means of numerical analysis, namely, the method of nume- rical integration of ordinary differential equations and the method of numerical continuation. We used toolbox Matcont [2], which works in the environment of program MATLAB [3]. The result is commented parametric study of the pheno- menon of corruption. Keywords: ordinary diferential equations, dynamic systems, bifurcation ana- lysis 1
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Methods for solving differential equations
Zadražil, Tomáš ; Staněk, Jakub (advisor) ; Slavík, Antonín (referee)
The goal of this work is to introduce some elementary methods of solving ordinary differential equations to the reader. These methods are then exemplified by means of using solved samples and particular illustrations from practice. The text intends to be friendly to an inexpert user, therefore it prefers verbal expressions to accumulation of mathematic expressions. The target groups are university students and secondary school teachers who teach optional seminars of mathematic. The knowledge of derivatives, definite and indefinite integrals is required to understand this text. Powered by TCPDF (www.tcpdf.org)
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Mathematical models of ecosystems
Scholle, David ; Janovský, Vladimír (advisor) ; Kofroň, Josef (referee)
This work is about models of population growth in different situations. At first, we will examine amount of spiders and their prey in the region of Langa Astigiana, based on models of dynamical systems. We will also consider the usage of spraying of near vineyards and effect of this on the ecosystem. The aim of this work is also to check the possibility of periodical cycles, and thus also of the Hopf Bifurcation, appearing. Next part talks about the model of a beehive and examines the influence of insecticides on the population of bee drones and worker bees. The aim of the last chapter is to examine the effectivity and possible impact of human intervention in the region of Šumava forest. The model will check the necessity of such action against parasites. The software used for these tasks will be mainly the continuation toolbox MatCont, which is a part of the program MatLab.
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An analysis of differential equations for systems involving bottlenecks
Borkovec, Ondřej ; Opluštil, Zdeněk (referee) ; Kisela, Tomáš (advisor)
This thesis deals with modelling of the flow of products through systems involving bottlenecks using ordinary differential equations. The model is based on hydrodynamics analogy. Further, the conditions for the sustainability of a system, that is the requirements needed not to exceed the maximal capacity, so that the flow of products can flow continuously through the given spot. A model is used to solve examples for vayrying systems.
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