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Course, academic year 2019/2020
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Differential Equations for Probability - NMTP462
Title in English: Diferenciální rovnice pro pravděpodobnost
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: summer
E-Credits: 3
Hours per week, examination: summer s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Guarantor: prof. RNDr. Bohdan Maslowski, DrSc.
Class: M Mgr. PMSE
M Mgr. PMSE > Volitelné
Classification: Mathematics > Probability and Statistics
Annotation -
Last update: T_KPMS (16.05.2013)
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.
Aim of the course -
Last update: T_KPMS (16.05.2013)

The subject is aimed at the study of certain parts of ordinary

differential equations and 2nd order partial differential equations both

of elliptic and parabolic types that are useful in probability theory.

Course completion requirements -
Last update: RNDr. Jitka Zichová, Dr. (25.04.2018)

Oral exam.

Literature - Czech
Last update: T_KPMS (16.05.2013)

J. Kurzweil: Obyčejné diferenciální rovnice. SNTL Praha, 1978.

A. Friedman: Partial Differential Equations of Parabolic Type. Prentice-Hall, N.J., 1964.

Teaching methods -
Last update: T_KPMS (16.05.2013)

Lecture.

Requirements to the exam
Last update: prof. RNDr. Bohdan Maslowski, DrSc. (09.10.2017)

The examination is oral.

Requirements (may be slightly modified according to the stuff talked over)::

Rigorous and correct formulations of all definitions and statements in the ODE part : generalized solution (in the Caratheodory sense), Thms. on local existence, nonexplosion, continuous dependence on initial data, definition of dynamical system, stability, asymptotic and exponential stability.

For the PDE part" Concept of solution to parabolic and elliptic PDE, fundamental solution, Green function, adjoint problem, maximum principles.

Proofs are required for Thm on local existence (based on Schauder fixed point thm), sufficient conditions for nonexplosion (T 1.6) incl. Corolary 1.7, and the theorems on stability (T 1.18, 1.19 and 1.20). Examples!

Syllabus -
Last update: T_KPMS (16.05.2013)

1) 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

2) the theory of linear partial differential equations; 1st order equations, the method of characteristics, classification of equations of the 2nd order, parabolic equations (the Cauchy problem, an outline of basic boundary value problems, the notion of Green function), elliptic equations (an outline of basic boundary value problems).

Entry requirements -
Last update: prof. RNDr. Bohdan Maslowski, DrSc. (24.05.2018)

In order to enroll, basic knowledge of standard calculus including measure theory and Lebesgue integral are required.

 
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