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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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Numerical solution of equations describing the dynamics of flocking
Živčáková, Andrea ; Kučera, Václav (advisor) ; Janovský, Vladimír (referee)
This work is devoted to the numerical solution of equations describing the dynamics of flocks of birds. Specifically, we pay attention to the Euler equations for compressible flow with a right-hand side correction. This model is based on the work Fornasier et al. (2010). Due to the complexity of the model, we focus only on the one-dimensional case. For the numerical solution we use a semi-implicit discontinuous Galerkin method. Discretization of the right-hand side is chosen so that we preserve the structure of the semi-implicit scheme for the Euler equations presented in the work Feistauer, Kučera (2007). The proposed numerical scheme was implemented and numerical experiments showing the robustness of the scheme were carried out. Powered by TCPDF (www.tcpdf.org)
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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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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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