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Course, academic year 2017/2018
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Selected Chapters on Partial Differential Equations - NMAF001
Czech title: Vybrané kapitoly z parciálních diferenciálních rovnic
Guaranteed by: Department of Geophysics (32-KG)
Faculty: Faculty of Mathematics and Physics
Actual: from 2003
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: Czech
Teaching methods: full-time
Guarantor: doc. RNDr. Ctirad Matyska, DrSc.
Classification: Physics > Mathematics for Physicists
Annotation -
Last update: T_KG (16.05.2001)

Classification of the equations of the second order. Weak formulation of the Dirichlet and the Neumann problem for the elliptic equations, mixed problems. Basic principles of the numerical solution - finite element method. Evolution equations.
Aim of the course -
Last update: T_KG (26.03.2008)

The lecture is devoted to introduction into the weak formulation of equations of mathematical physics and their numerical solution.

Literature - Czech
Last update: RNDr. Pavel Zakouřil, Ph.D. (05.08.2002)

  • K. Rektorys: Variační metody v inženýrských problémech matematické fyziky, SNTL, Praha 1974.
  • M. Křížek, P. Neittaanmaki: Finite Element Approximation of Variational Problems and Applications, Longman and J. Wiley & Sons, New York, 1990.

Teaching methods -
Last update: T_KG (11.04.2008)

Lecture

Syllabus -
Last update: T_KG (07.05.2002)

Introductory concepts

Classical solutions, domains with the Lipschitz boundary, Green's theorem, classification of the equations of the second order, Fourier method demonstrated on the scalar wave equation.

Sobolev spaces

Definition of the Sobolev space W1,2 , trace theorem, Rellich's theorem.

Linear elliptic equations - weak and variational formulations

Dirichlet's problem - formulation and interpretation of the weak solution; Lax-Milgram theorem and uniqueness of the problem; variational approach - differentiating in the Gateaux sense of the functional of potential energy; sufficient conditions for the existence of the minimum; generalized problem for elliptic equations - existence and uniqueness, Neumann's problem and equilibrium conditions.

Nonlinear equations

Strictly monotone operators and contraction theorem, uniqueness of the solution.

Spectral theory

Definition and properties of the Green operator; eigenvalues of the Green operator.

Finite elements

Basic concepts and ideas of the finite element method. Numerical examples.

Literature:

M. Křížek, P. Neittaanmaki: Finite Element Approach of Variational Problems and Applications, Longman and J. Wiley & Sons, New York, 1990.

 
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