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Convex hull properties for parabolic systems of partial differential equations
Češík, Antonín ; Schwarzacher, Sebastian (advisor) ; Bulíček, Miroslav (referee)
The topic of this thesis is the convex hull property for systems of partial differential equations, which is a natural generalisation of the maximum principle for scalar equations. The main result of this thesis is a theorem asserting the convex hull property for the solutions of a certain class of parabolic systems of nonlinear partial differential equations. It also investigates the coefficients of linear systems. The respective results are sharp which is demonstrated by counterexamples to the convex hull property for solutions of linear elliptic and parabolic systems. The general theme is that the coupling of the system is what breaks the convex hull property, not necessarily the non-linearity.
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Modelica in physiological modelling. Models with spatially distributed parameters, Authorin educational simulators.
Šilar, Jan ; Kofránek, Jiří (advisor) ; Maršálek, Petr (referee) ; Roubík, Karel (referee)
Mathematical models in physiology are useful to formulate and verify hypotheses, to make predictions, to estimate hidden parameters and in education. This thesis deals with modelling in physiology using the Modelica language. New methods for model implementation and simulator production were developed. Modelica is an open standard equation-based object-oriented language for modelling complex systems. It is highly convenient in physiology modelling due to its ability to describe extensive models in a lucid hierarchical way. The models are described by algebraic, ordinary differential and discrete equations. Partial differential equations are not supported by the Modelica standard yet. The thesis focuses on two main topics: 1) modelling of systems described by partial differential equations in Modelica 2) production of web-based e-learning simulators driven by models implemented in Modelica. A Modelica language extension called PDEModelica1 for 1-dimensional partial differential equations was designed (based on a previous extension). The OpenModelica modelling tool was extended to support PDEModelica1 using the method of lines. A model of countercurrent heat exchange between the artery and vein in a leg of a bird standing in water was implemented using PDEModelica1 to prove its usability. The...
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Modelling of bioelectronic devices
Truksa, Jan ; Vala, Martin (referee) ; Salyk, Ota (advisor)
Tématem této práce je počítačové modelování organického elektrochemického tranzistoru (OECT). Pro vytvoření modelu bylo třeba vypočítat rozložení elektrického pole a koncentrace iontů elektrolytu. Výpočet byl proveden numericky pomocí metody konečných prvků. Bylo vypočítáno rozložení elektrického potenciálu na povrchu kanálu OECT, dále byly vypočítány změny vodivosti a výstupní proud OECT. Výpočty byly provedeny na osobním počítači pomocí komerčního softwaru COMSOL Multiphysics. Kvůli nedostatečnému výpočetnímu výkonu musel být model rozdělen na části a drasticky zjednodušen. Prezentované výsledky se liší od literatury, protože se nepodařilo správně modelovat saturaci tranzistoru. Odchylky od reálného chování OECT jsou pravděpodobně způsobeny zjednodušením modelu.
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Adaptive methods for singularly perturbed partial differential equations
Lamač, Jan ; Knobloch, Petr (advisor)
This thesis deals with solving singularly perturbed convection- diffusion equations. Firstly, we construct a matched asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in 1D and derive a formula for the zeroth-order asymptotic expansion in several two- dimensional polygonal domains. Further, we present a set of stabilization meth- ods for solving singularly perturbed problems and prove the uniform convergence of the Il'in-Allen-Southwell scheme in 1D. Finally, we introduce a modification of the streamline upwind Petrov/Galerkin (SUPG) method on convection-oriented meshes. This new method enjoys several profitable properties such as the ful- filment of the discrete maximum principle. Besides the analysis of the method and derivation of a priori error estimates in respective energy norms we also carry out several numerical experiments verifying the theoretical results.
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Adaptive methods for singularly perturbed partial differential equations
Lamač, Jan ; Knobloch, Petr (advisor) ; Franz, Sebastian (referee) ; Vejchodský, Tomáš (referee)
This thesis deals with solving singularly perturbed convection- diffusion equations. Firstly, we construct a matched asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in 1D and derive a formula for the zeroth-order asymptotic expansion in several two- dimensional polygonal domains. Further, we present a set of stabilization meth- ods for solving singularly perturbed problems and prove the uniform convergence of the Il'in-Allen-Southwell scheme in 1D. Finally, we introduce a modification of the streamline upwind Petrov/Galerkin (SUPG) method on convection-oriented meshes. This new method enjoys several profitable properties such as the ful- filment of the discrete maximum principle. Besides the analysis of the method and derivation of a priori error estimates in respective energy norms we also carry out several numerical experiments verifying the theoretical results.
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Computer modeling of the inner ear
Perlácová, Tereza ; Jungwirth, Pavel (advisor) ; Vejchodský, Tomáš (referee)
Do mechanického modelu kochley zavádzame implicitné numerické metódy. Tes- tujeme konkrétne štyri metódy: implicitný Euler, Crank-Nicolson, BDF druhého a tretieho rádu na lineárnej a nelineárnej verzii modelu. Nelineárny model obsahuje funkciu so saturujúcou vlastnosťou. Aplikácia implicitných metód na nelineárny model vedie na sústavu nelineárnych rovníc. Predstavujeme dva spôsoby, ako túto sústavu numericky riešiť. Prvý z nich zahrňuje nelinearitu do pravej strany novovzniknutej lineárnej sústavy. Druhý robí linearizáciu nelineárnej funkcie. V práci porovnávame oba spôsoby z hľadiska efektivity a sledujeme ich konvergenciu k referenčnému riešeniu. Pre hodnotu tolerancie, ktorú používame na určenie numerickej konvergencie, je prvý spôsob efektívnejší. V úplne nelineárnom režime druhý spôsob zlyháva, pretože nekon- verguje k referenčnému riešeniu. Výsledkom porovnania implicitných metód je, že Crank-Nicolsonova metóda s prvým spôsobom riešenia nelineárnej sústavy je pre účely nášho modelu najlepšia. Použitie tejto metódy v mechanickom modeli nám umožňuje vytvoriť ľubovoľne presné prepojenie medzi mechanickým a elektrickým modelom kochley, rešpektujúc fyziológiu človeka. 1
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Analysis of Methods of Differences for Partial Differential Equations Solving
Zpěváková, Jana ; Zbořil, František (referee) ; Šátek, Václav (advisor)
In this thesis, we discuss the numerical solution of ordinary differential equation and numerical methods of solving partial differential equations. We propose and implement an application, that converts partial differential hyperbolic equation to a set of ordinary differential equations using finite difference method. After that, the system of equations is solved using the Taylor method programmed in Matlab environment. Finally, we compare the time complexity of proposed solution with parallel numerical computation.
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Bifurcation in mathematical models in biology
Kozák, Michal ; Stará, Jana (referee)
Stationary, spatially inhomogenous solutions of reaction-diffusion systems are studied in this thesis. These systems appears in biological models based on a Tu- ring's idea of a diffusion driven instability. In the connection, a global behaviour of bifurcation branches of these stationary solutions is analyzed. The thesis in- sists on theory of differential equations and on (particularly topological) methods of nonlinear analysis. The existence, as well as non-compatness in one-dimensional space, of a bifurcation branch of general reaction-diffusion system leading to Tu- ring's efekt is proved. Further, a priori estimates of Thomas model are derived. The results tend to theorem, that forall diffusion coefficient from the preestab- lished set there exists at least one stacionary, spacially nontrivial solution of Tho- mas model.
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Bifurcation in mathematical models in biology
Kozák, Michal ; Kučera, Milan (advisor) ; Stará, Jana (referee)
Stationary, spatially inhomogenous solutions of reaction-diffusion systems are studied in this thesis. These systems appears in biological models based on a Tu- ring's idea of a diffusion driven instability. In the connection, a global behaviour of bifurcation branches of these stationary solutions is analyzed. The thesis in- sists on theory of differential equations and on (particularly topological) methods of nonlinear analysis. The existence, as well as non-compatness in one-dimensional space, of a bifurcation branch of general reaction-diffusion system leading to Tu- ring's efekt is proved. Further, a priori estimates of Thomas model are derived. The results tend to theorem, that forall diffusion coefficient from the preestab- lished set there exists at least one stacionary, spacially nontrivial solution of Tho- mas model.
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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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