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Stiff Systems Analysis
Šátek, Václav ; Dalík, Josef (referee) ; Horová, Ivana (referee) ; Kunovský, Jiří (advisor)
The solving of stiff systems is still a contemporary sophisticated problem. The basic problem is the absence of precise definition of stiff systems. A question is also how to detect the stiffness in a given system of differential equations. Implicit numerical methods are commonly used for solving stiff systems. The stability domains of these methods are relatively large but the order of them is low. The thesis deals with numerical solution of ordinary differential equations, especially numerical calculations using Taylor series methods. The source of stiffness is analyzed and the possibility how to reduce stiffness in systems of ordinary differential equations (ODEs) is introduced. The possibility of detection stiff systems using explicit Taylor series terms is analyzed. The stability domains of explicit and implicit Taylor series are presented. The solutions of stiff systems using implicit Taylor series method are presented in many examples. The multiple arithmetic must be used in many cases. The new suitable parallel algorithm based on implicit Taylor series method with recurrent calculation of Taylor series terms and Newton iteration method (ITMRN) is proposed.
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Platform .NET for Numerical Integration
Kopecký, Jiří ; Kunovský, Jiří (referee) ; Šátek, Václav (advisor)
This bachelor thesis deals with numeric solution of ordinary first-order differential equations and their systems. The first part of this thesis contains description of selected one-step integration methods. The second part is devoted to a language intended for differential equation notation. This part at first describes the study of languages of MATLAB, Maple and TKSL/386 simulation systems. Later, based on this study it presents a design of a new language. The penultimate part of the thesis deals with design and implementation of a system intended for the calculation of systems of differential equations. In the final part is then shown usage of this system to solve exercises from the Circuits Theory domain.
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Stiff Systems Analysis
Šátek, Václav ; Dalík, Josef (referee) ; Horová, Ivana (referee) ; Kunovský, Jiří (advisor)
The solving of stiff systems is still a contemporary sophisticated problem. The basic problem is the absence of precise definition of stiff systems. A question is also how to detect the stiffness in a given system of differential equations. Implicit numerical methods are commonly used for solving stiff systems. The stability domains of these methods are relatively large but the order of them is low. The thesis deals with numerical solution of ordinary differential equations, especially numerical calculations using Taylor series methods. The source of stiffness is analyzed and the possibility how to reduce stiffness in systems of ordinary differential equations (ODEs) is introduced. The possibility of detection stiff systems using explicit Taylor series terms is analyzed. The stability domains of explicit and implicit Taylor series are presented. The solutions of stiff systems using implicit Taylor series method are presented in many examples. The multiple arithmetic must be used in many cases. The new suitable parallel algorithm based on implicit Taylor series method with recurrent calculation of Taylor series terms and Newton iteration method (ITMRN) is proposed.
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Platform .NET for Numerical Integration
Kopecký, Jiří ; Kunovský, Jiří (referee) ; Šátek, Václav (advisor)
This bachelor thesis deals with numeric solution of ordinary first-order differential equations and their systems. The first part of this thesis contains description of selected one-step integration methods. The second part is devoted to a language intended for differential equation notation. This part at first describes the study of languages of MATLAB, Maple and TKSL/386 simulation systems. Later, based on this study it presents a design of a new language. The penultimate part of the thesis deals with design and implementation of a system intended for the calculation of systems of differential equations. In the final part is then shown usage of this system to solve exercises from the Circuits Theory domain.
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