Mathematics PS

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

After elementary tools of linear algebra (matrix, determinant, Gaussian elimination) are recalled, iterative methods for solving systems of linear algebraic equations are in the focus. Then, the finite difference method and the finite element method are presented and their applications to problems based on differential equations are shown.

Requirements:

At least 70% excercise class attendance.

Syllabus of lectures:

1. Matrix, inner product of vectors, eigenvalues and eigenvectors of a matrix, spectrum of a matrix, Gershgorin theorem.

2. Normed linear space, matrix and vector norms, condition number.

3. Iterative methods for solving systems of linear algebraic equations, sparse matrix.

4. 0rdinary differential equation with boundary conditions, eigenvalue and eigenfunction problem.

5. Function spaces, inner product of functions, differential operators.

6. Variational principle in 1D problems with positive definite operators, functional of energy, generalized (weak) solution.

7. Approximate solution obtained by variational methods (Ritz method and the finite element method in 1D).

8. Poisson equation in 2D, boundary conditions, applications, Ritz method, finite element method.

9. The finite difference method for 1D boundary value problems and eigenvalue/eigenfuncion problems. Different boundary conditions.

10. Finite difference method for 2D elliptic boundary value problems, Liebmann iteration ( just for information).

11.Wave equation, numerical solution by the finite difference method, stable and unstable method.

12. Transient heat equation, numerical solution by the finite difference method (in 2D - just for information), stable and unstable method.

13. Reserve

Syllabus of tutorials:

Exercise classes follow the topics of lectures.

Study Objective:

Students wil become familiar with basic tools and numerical methods for solving common problems based on differential equations.

Study materials:

Educational materials on the 101MA04 website, e.g., J. Chleboun: Příklady k předmětu Matematika 4; J. Chleboun: Matematika 4 - příručka pro přežití; J. Chleboun: PDF presentations

O. Zindulka: Matematika 3, Chapter 4, 5, 6; Česká technika - nakladatelství ČVUT, Praha, 2007

K. Rektorys: Matematika 43,Česká technika - nakladatelství ČVUT, Praha, 2001

Note:

Further information:

Viz https://mat.fsv.cvut.cz/vyuka/magistri/zs

No time-table has been prepared for this course

The course is a part of the following study plans:

• Stavební inženýrství, obor Konstrukce pozemních staveb (compulsory course)
• obor Konstrukce a dopravní stavby, zaměření Inženýrské konstrukce (compulsory course)
• obor Konstrukce a dopravní stavby, zaměření Dopravní stavby (compulsory course)