Partial Differential Equations I

Garant předmětu:

Lecturer:

Tutor:

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:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• 13 136 NSTI MMT 2012 základ (compulsory course in the program)