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A traffic flow with a bottelneck
Kovařík, Adam ; Janovský, Vladimír (advisor) ; Vejchodský, Tomáš (referee)
Title: A traffic flow with a bottelneck Author: Adam Kovařík Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Vladimír Janovský, DrSc. Supervisor's e-mail address: janovsky@karlin.mff.cuni.cz Abstract: In this paper we study a microscopic follow-the-leader traffic model on a circu- lar road with a bottleneck. We assume that all drivers are identical and overtaking is not permitted. We sketch a small part of the rich dynamics of the model including Hopf and Neimark-Sacker bifurcations. We introduce so called POM and quasi-POM solutions and an algorithm how to search them. The main goal of this work is to investigate how the optimal velocity model with a bottleneck deals with so called aggressive behavior of dri- vers. The effect of variable reaction time and a combination of both named factors is also tested. Using numerical simulations we'll find out that aggressiveness and faster reactions have positive effect on traffic flow. In the end we discuss models with two bottlenecks and with one extraordinary driver. Keywords: dynamical systems, ODEs, traffic flow, bottleneck, aggressiveness. 1
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Application of the Laplace transoform and the homotopy perturbation method for the Burgers equation
Chaloupka, Tomáš ; Felcman, Jiří (advisor) ; Janovský, Vladimír (referee)
We use the homotopy perturbation method for solving different types of functional equations. The method is formulated in Introduction. Several types of functional equations are solved in Chapter one. In Chapter two, we define the Laplace transformation and combine it with the homotopy perturbation method in order to solve some differential equations. Last chapter is focused on attempts to find the solution of the Burgers equation with different initial conditions. For these conditions, we try to prove the existence of the solution or to find a suitable approximation of the solution. We compared the method with the method of characteristics. We investigate the behaviour of the homotopy perturbation method where method of characteristics doesn't exclude the existence of the classic solution. We discuss the practical application of the homotopy perturbation method to the Burgers equation.
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A quadrature formula of Clenshaw-Curtis type for the Gegenbauer weight-function
Labant, Ján ; Kofroň, Josef (advisor) ; Janovský, Vladimír (referee)
In this thesis we study especially quadrature formulae based on the Cheby- shev expansion, known as the Clenshaw-Curtis quadrature. The first part is focused on the Chebyshev polynomials, their definitions and properties. This knowledge will be used to derivate the Clenshaw-Curtis quadrature. Consider- able part of this work is dedicated to comparison of this and the well-known Gauss quadrature both theoretically and practicaly. In the further work we will extend the Clenshaw-Curtis quadrature by the Gegenbauer weight function which gives us new methods for numerical integration. These methods allow us to find a solution of some known problems what will be pointed out also on some nu- merical experimets. 1
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Oscillations in mechanical systems with implicit constitutive relations.
Babováková, Jana ; Pražák, Dalibor (advisor) ; Janovský, Vladimír (referee)
We study a system of differential-algebraic equations, describing motions of a mass-spring-dashpot oscillator by three different forms of implicit constitu- tive relations. For some problems with fully implicit but linear constitutive laws for combined force, we find conditions for solution stability. Assuming monotone relationship between the displacement, velocity and the respective forces, we prove global existence of the solutions. For a linear spring and a dashpot with maximal monotone relationship between the damping force and the velocity, we prove the global existence and uniqueness result. We also solve this problem numerically for Coulomb-like damping term.
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Analýza výpočtu největšího společného dělitele polynomů
Kuřátko, Jan ; Zítko, Jan (advisor) ; Janovský, Vladimír (referee)
In this work, the analysis of the computation of the greatest common divisor of univariate and bivariate polynomials is presented. The whole process is split into three stages. In the first stage, data preprocessing is explained and the resulting better numerical behavior is demonstrated. Next stage is concerned with the problem of the computation of the numerical rank of the Sylvester matrix, from which the degree of the greatest common divisor is obtained. The last stage is the actual algorithm for calculating the greatest common divisor of two polynomials. Furthermore, the underlying theory behind the computation of the greatest common divisor is explained and illustrated on many examples. 1
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Lineární algebraické modelování úloh s nepřesnými daty
Vasilík, Kamil ; Hnětynková, Iveta (advisor) ; Janovský, Vladimír (referee)
In this thesis we consider problems Ax b arising from the discretization of ill-posed problems, where the right-hand side b is polluted by (unknown) noise. It was shown in [29] that under some natural assumptions, using the Golub-Kahan iterative bidiagonalization the noise level in the data can be estimated at a negligible cost. Such information can be further used in solving ill-posed problems. Here we suggest criteria for detecting the noise revealing iteration in the Golub-Kahan iterative bidiagonalization. We discuss the presence of noise of different colors. We study how the loss of orthogonality affects the noise revealing property of the bidiagonalization.
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Solving bordered linear systems
Štrausová, Jitka ; Janovský, Vladimír (advisor) ; Zítko, Jan (referee)
The comparison of two algorithms for solving bordered linear systems is considered. The matrix of this system consists of four blocks (matrices A,B,C,D), the upper left one is a sparse matrix A, which is ill-conditioned and structured. The other blocks (B,C,D) are dense. We say that the matrix A is bordered with the matrices B,C,D. It is desirable to preserve the block structure of the matrix and take advantage of sparsity and structure of the matrix A. The literature suggests to use two different algorithms: The first one is the method BEM for matrices with the borders of width equal to one. The recursive alternative for matrices with wider borders is called BEMW. The second algorithm is an iterative method. Both techniques are based on different variants of the block LU-decomposition.
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