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CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2019/2020

Partial Differential Equations I

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Code Completion Credits Range Language
2011088 ZK 5 2P+1C Czech
Lecturer:
Stanislav Kračmar (guarantor)
Tutor:
Stanislav Kračmar (guarantor)
Supervisor:
Department of Technical Mathematics
Synopsis:

The course contains the essential parts of the classical theory of partial differential equations (PDE), first-order equations, the classification of second-order equations, the derivation of some important equations of mathematical physics, the method of characteristics, the Fourier method of the series. The theory of elliptical equations, principles of maxima, the uniqueness of solutions, potential methods, the concept of a fundamental solution and the method of the Green functions will be discussed in more detail.

Students will be acquainted with the apparatus used in the field of partial differential equations: Fourier transform and its use. Distributions and generalized derivatives. Important inequalities: Friedrich's inequality, Poincare's inequality, Minkowsky inequality,

Mathematical means used in the so-called modern PDE theory will be discussed, the basis of which will be the subject of PDE II: Fundamentals of functional analysis: Hilbert spaces, Banach spaces, and their properties, linear operators in these spaces. Riesz's theorem. The concept of the continuous embedding and the compact embedding. Convergent and weakly convergent sequences. Sobolev spaces, the theorem on the equivalence of norms, the theorem on traces of functions from Sobolev's space, assertions on continuous and compact embeddings of Sobolev spaces.

Introduction in variational methods of PDE. Using the results of the functional analysis to introduce and study weak solutions of elliptic, parabolic and hyperbolic equations.

Requirements:
Syllabus of lectures:

The course contains the essential parts of the classical theory of partial differential equations (PDE), first-order equations, the classification of second-order equations, the derivation of some important equations of mathematical physics, the method of characteristics, the Fourier method of the series. The theory of elliptical equations, principles of maxima, the uniqueness of solutions, potential methods, the concept of a fundamental solution and the method of the Green functions will be discussed in more detail.

Students will be acquainted with the apparatus used in the field of partial differential equations: Fourier transform and its use. Distributions and generalized derivatives. Important inequalities: Friedrich's inequality, Poincare's inequality, Minkowsky inequality,

Mathematical means used in the so-called modern PDE theory will be discussed, the basis of which will be the subject of PDE II: Fundamentals of functional analysis: Hilbert spaces, Banach spaces, and their properties, linear operators in these spaces. Riesz's theorem. The concept of the continuous embedding and the compact embedding. Convergent and weakly convergent sequences. Sobolev spaces, the theorem on the equivalence of norms, the theorem on traces of functions from Sobolev's space, assertions on continuous and compact embeddings of Sobolev spaces.

Introduction in variational methods of PDE. Using the results of the functional analysis to introduce and study weak solutions of elliptic, parabolic and hyperbolic equations.

Syllabus of tutorials:
Study Objective:
Study materials:

K. Rektorys: Variational Methods in Mathematics, Science and Engineering. Springer, Netherlands,1977.

L. C. Evans: Partial differential equations. Graduate Studies in Mathematics, Vol 19, American Mathematical Society, Second Edition 2010.

M. Chipot: Elliptic equations. An Introductory cource. Birkhauser Verlag, 2009.

M. E. Taylor: Partial differential equations I. Applied Mathematical Sciences 115, Springer, 2011.

Note:
Time-table for winter semester 2019/2020:
06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue
Fri
roomKN:D-105
Kračmar S.
14:15–15:45
(lecture parallel1)
Karlovo nám.
Konzultační místnost 12101
roomKN:D-105
Kračmar S.
16:00–16:45
(lecture parallel1
parallel nr.101)

Karlovo nám.
Konzultační místnost 12101
Thu
Fri
Time-table for summer semester 2019/2020:
Time-table is not available yet
The course is a part of the following study plans:
Data valid to 2019-09-16
For updated information see http://bilakniha.cvut.cz/en/predmet1896406.html