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Fundamental numerical schemes for fractional differential equations
Strouhal, Jiří ; Nechvátal, Luděk (referee) ; Tomášek, Petr (advisor)
This bachelor's thesis focuses on fractional calculus, fractional differential equations with the Caputo derivative, and primarily on numerical methods for solving these equations. Numerical methods play a crucial role for this type of equations, as only a small fraction have known analytical solutions, requiring dependence on numerical solutions. This thesis will cover the following numerical methods: explicit and implicit Euler methods, and the predictor-corrector method. The objective is to implement selected methods in MATLAB and test them on several initial problems.
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Numerical methods in economics
HRNČÍŘ, Pavel
The bachelor thesis focuses on the use of numerical methods in economic practice. The thesis describes the essence of interpolation and regression. Different methods of function approximation are presented. The aim of the thesis is to evaluate whether selected numerical methods, namely interpolation polynomial and linear regression, are suitable for predicting future values of economic variables based on historical data and whether interpolation polynomial is suitable for estimating missing values among known data. The economic variables worked with are the average annual inflation rate and the average gross monthly wage. The data of the economic variables are obtained from the Czech Statistical Office. A console application in the C# programming language has been developed for this work, which allows the calculation of the interpolation polynomial from the given data.
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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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Algebraic Equations Solution Convergence
Sehnalová, Pavla ; Šátek, Václav (referee) ; Kunovský, Jiří (advisor)
The work describes techniques for solving systems of linear and differential equations. It explains the definition of conversion from system of linear to system of differential equations. The method of the elementary transmission and the transform algorithm are presented. Both of methods are demonstrated on simply examples and properties of conversion are shown. The work compares fast and accurate solutions of methods and algorithm. For computing examples and solving experiments following programs were used: TKSL and TKSL/C. The program TKSL/C was enriched with the graphic user interface which makes the conversion of systems and computing results easier.
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Design and analysis of membrane roof of a stadium.
Lang, Rostislav ; Kytýr,, Jií (referee) ; Němec, Ivan (advisor)
This diploma thesis deals with problem of design and calculation of membrane structure of stadium roof. This is a complex engineering problem, which includes many partial problems: finding of initial form of membrane, statically and architecturally suitable arrangement of catenaries, economical solution of boundary conditions (foundations). All components affect each other and cannot be dealt without mutual coordination. It always greatly depends on the experience and intuition of engineer who design such structure. Task which cannot be resolved according to the theory of the first order. Equilibrium forces on the deformed structure, which in many projected structures gives satisfactory results, did not correspond to reality. It is therefore necessary to consider equilibrium of forces on the deformed structure according to the theory of large deformations. Diploma thesis was entered with regard to the intention of the companies Ing. Software Dlubal s.r.o. and FEM consulting s.r.o., working together to develop software RFEM. These companies plan to complement this program system with a module MEMBRANE for searching of initial shapes of membrane structures. This work is a contribution to the creation of this module.
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Image analysis for correction of electron microscopes
Smital, Petr ; Schwarz, Daniel (referee) ; Kolář, Radim (advisor)
This thesis describes the physical nature of corrections of an electron microscope and mathematical methods of image processing required for their complete automation. The corrections include different types of focusing, astigmatism correction, electron beam centring, and image stabilisation. The mathematical methods described in this thesis include various methods of measuring focus and astigmatism, with and without using the Fourier transform, edge detection, histogram operations, and image registration, i.e. detection of spatial transformations in images. This thesis includes detailed descriptions of the mathematical methods, their evaluation using an “offline” application, descriptions of the algorithms of their implementation into an actual electron microscope and results of their testing on the actual electron microscope, in the form of a video footage grabbed from its control computer’s screen.
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Acceleration of numerical computation of heat conduction in solids in inverse tasks
Ondruch, Tomáš ; Komínek, Jan (referee) ; Pohanka, Michal (advisor)
The master's thesis deals with possible ways of accelerating numerical computations, which are present in problems related to heat conduction in solids. The thesis summarizes basic characteristics of heat transfer phenomena with emphasis on heat conduction. Theoretical principles of control volume method are utilized to convert a direct heat conduction problem into a sparse linear system. Relevant fundamentals from the field of inverse heat conduction problems are presented with reference to intensive computations of direct problems of such kind. Numerical methods which are well-suited to find a solution of direct heat conduction problems are described. Remarks on practical implementation of time-efficient computations are made in relation with a two-dimensional heat conduction model. The results are compared and discussed with respect to obtained computational time for several tested methods.
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