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In the lecture, some selected chapters from the theory of differential equations are dealt with. In particular, in the theory of ordinary differential equations: the notion of Caratheodory solution and its existence and uniqueness, continuous dependence on the initial datum, linear equations in a Euclidean space-structure of solutions, the fundamental matrix, variation of constants; in the theory of linear partial differential equations: 1st order equations, the method of characteristics, classification of equations of the 2nd order,
parabolic equations, elliptic equations.
Last update: Zichová Jitka, RNDr., Dr. (20.05.2026)
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The subject is aimed at the study of certain parts of ordinary differential equations, the heat equation and parabolic partial differential equations that are useful in probability theory. Last update: Slavík Jakub, Mgr., Ph.D. (20.05.2026)
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Oral exam. Last update: Zichová Jitka, RNDr., Dr. (25.04.2018)
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J. Kurzweil: Obyčejné diferenciální rovnice. SNTL Praha, 1978.
A. Friedman: Partial Differential Equations of Parabolic Type. Prentice-Hall, N.J., 1964. Last update: T_KPMS (16.05.2013)
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Lecture. Last update: T_KPMS (16.05.2013)
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The examination is oral. Requirements (may be modified based on current material):
Rigorous and correct formulations of all definitions and statements: ODE part: generalized solution (in the Carathéodory sense), theorems on local existence, nonexplosion, continuous dependence on initial data, definition of dynamical system, stability, asymptotic and exponential stability. PDE part: concept of solution of heat equation and parabolic PDEs, fundamental solution, existence theorem for heat equation and parabolic PDEs, uniqueness for heat equation. Examples.
Proofs required: ODE part: local existence (based on Schauder fixed point), extension to maximal solutions, sufficient conditions for nonexplosion, theorems on stability. PDE part: existence of solutions of heat equation. Last update: Slavík Jakub, Mgr., Ph.D. (20.05.2026)
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1) the theory of ordinary differential equations; the notion of Carathéodory solution and its existence and uniqueness, continuous dependence on the initial datum, linear equations in a Euclidean space-structure of solutions, the fundamental matrix, variation of constants, indirect and direct Lyapunov stability.
2) the theory of linear partial differential equations; classification of PDEs, heat equation (the Cauchy problem), parabolic equations. Last update: Slavík Jakub, Mgr., Ph.D. (20.05.2026)
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In order to enroll, basic knowledge of standard calculus including measure theory and Lebesgue integral are required. Last update: Maslowski Bohdan, prof. RNDr., DrSc. (24.05.2018)
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