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Numerical methods of solving fractional differential equations
Kyjovský, Adam ; Zatočilová, Jitka (referee) ; Nechvátal, Luděk (advisor)
This bachelor's thesis deals with numerical methods of solving fractional differential equations. Some fundamental notions of fractional calculus and basic results from the theory of fractional differential equations (such as existence and uniqueness of the solution to an initial value problem with the Caputo derivative) are presented. Further, a summary of selected numerical methods for solving such initial value problems is presented. These methods are tested and compared on a model problem.
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Runge-Kutta methods
Kroulíková, Tereza ; Nechvátal, Luděk (referee) ; Zatočilová, Jitka (advisor)
Tato práce se zabývá Runge--Kuttovými metodami pro počáteční problém. Práce začíná analýzou Eulerovy metody a odvozením podmínek řádu. Jsou představeny modifikované metody. Pro dvě z nich je určen jejich řád teoreticky a pro všechny je provedeno numerické testování řádu. Jsou představeny a numericky testovány dva typy metod s odhadem chyby, "embedded" metody a metody založené na modifikovaných metodách. V druhé části jsou odvozeny implicitní metody. Jsou představeny dva způsoby konstrukce implicitních "embedded" metod. Jsou zmíněny také diagonální implicitní metody. Na závěr jsou probrány dva druhy stability u metod prezentovaných v práci.
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Time discretization method of solving PDE
Myška, Michal ; Zatočilová, Jitka (referee) ; Nechvátal, Luděk (advisor)
This thesis deals with solving evulution partial differential equations by the time discretization method. It originates form the Rothe's method (methond of lines). The thesis is divided into three parts. The first one shows principle of the method. The second part focuses on teoretical aspects, in particular, on existence and convergence theorem along with an error estimate. Some function analysis tools are presented here as well. In the last part, a MATLAB code is listed.
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Newton and numerical mathematics
Obrátil, Štěpán ; Nechvátal, Luděk (referee) ; Zatočilová, Jitka (advisor)
Topic of this bachelor thesis are Newton's methods for numerical solutions of various problems. Especially the problems of solving nonlinear equations and systems of nonlinear equations, as well as numerical integration are explained. The Newton's method for solving nonlinear equations is presented, as well as its many modifications and its generalisation for systems of nonlinear equations. Usefulness of methods is demonstrated on various examples. In the end, Newton-Cotes quadrature formulae for numerical integration are presented.
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Numerical solution of index-2 differenial-algebraic equations
Kroulíková, Tereza ; Opluštil, Zdeněk (referee) ; Zatočilová, Jitka (advisor)
This bachelor´s thesis deals with numerical solution of differential-algebraic equations. At first these equations are described theoretically and their basic properties are presented. Main attention is paid to index and the most used indexes are described in details. Then the thesis concentrates on numerical solution of Hessenberg forms index-2 differential-algebraic equations. Implicit Runge-Kutta methods and backward differentiation formulas are derived. Those are used for solution of index-2 differential-algebraic equations.
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Numerical simulation of sound propagation by difference methods
Prochazková, Zdeňka ; Zatočilová, Jitka (referee) ; Čermák, Libor (advisor)
The goal of this thesis is to introduce the finite difference method (FDM) adjusted for usage in modeling of sound propagation, and other approaches that are used together with this method. These approaches include selective filtering and time integration using the Runge-Kutta method, which has low computer memory requirements. An important topic in modeling sound propagation are boundary conditions. The thesis examines and verifies several types of boundary conditions. Included in the thesis are solutions to example problems implemented in Matlab.
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Meshless methods for computational fluid dynamics
Niedoba, Pavel ; Zatočilová, Jitka (referee) ; Čermák, Libor (advisor)
The thesis deals with meshfree methods, especially the SPH method. It is focused on the question of convergence near the boundary of the problem domain and its following solution in the form of using the so-called ghost particles as a boundary condition. There is also presented a suitable setting of parameters for a shock tube 2D problem based on many tests and software modifications.
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Modelling and simulation in aviation
Prešinský, Ján ; Nechvátal, Luděk (referee) ; Zatočilová, Jitka (advisor)
This bachelor thesis is focused on specifying the orientation of aircraft in normal Earth- xed frame. It is devoted to Euler angles representation and quaternions representation. Moreover, it introduces the equations of motion with 3 and 6 degrees of freedom and proposes the numerical method for solving these equations, which are expressed by the set of non-linear diferential equations.
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