Mathematics 4B

Garant předmětu:

Ondřej Zindulka

Lecturer:

Jan Chleboun, Yuliya Namlyeyeva, Ondřej Zindulka

Tutor:

Jan Chleboun, Yuliya Namlyeyeva, Ondřej Zindulka

Supervisor:

Department of Mathematics

Synopsis:

Spectral theory of matrices and linear operators. Boundary value problems, in particular those modeling the beam. Variational principle, variational methods.

Requirements:

1. Eigenvalues, spectrum of a matrix, spectral radius, Gershgorin Theorem; norms; symmetric and positive definite matrices; Cholesky factorization; condition number; energy and variational principle; convergence of Steepest Descent and Conjugate Gradients methods.

2. Linear differential equations; null space, fundamental system, Wronski determinant, reduction of order; Cauchy problems, variation of constants; equations with constant coefficients (also higher order)

3. Boundary value problems; eigenvalues, eigenfunctions, eigenspaces; solvability. Linear differential operators; bounded, positive and weakly positive operators; eigenvalues and eigenfunctions; Fredholm Theorem, solvability.

4. Energy of Au = f. Variational principle. Stability.

Syllabus of lectures:

1. Matrices, eigenvalues, eigenvectors.

2. Norms of matrices and vectors. Large systems of linear equations.

3. Symmetric and positive-definite matrices. Condition number. Direct methods of linear equations systems solution.

4. Variational principle for linear equations systems. Iterative methods of linear equations systems solution. The conjugate gradient method as a direct and iterative method.

5. Initial value problem for ordinary differential equation. Fundamental system and general solution.

6. Dot product and orthogonality of continuous functions. Boundary value problem. Eigenvalues and eigenfunctions of a boundary value problem. 7. Existence and uniqueness of a solution.

8. The energy of an equation. Variational principle.

9. Application: The deflection of simply supported and clamped beams.

10. Energy and stability.

11. Selected partial differential equations: Laplace, heat and wave equations.

12. Selected partial differential equations: Laplace, heat and wave equations.

Syllabus of tutorials:

1. Matrices, eigenvalues, eigenvectors.

2. Norms of matrices and vectors. Large systems of linear equations.

3. Symmetric and positive-definite matrices. Condition number. Direct methods of linear equations systems solution.

4. Variational principle for linear equations systems. Iterative methods of linear equations systems solution. The conjugate gradient method as a direct and iterative method.

5. Initial value problem for ordinary differential equation. Fundamental system and general solution.

6. Dot product and orthogonality of continuous functions. Boundary value problem. Eigenvalues and eigenfunctions of a boundary value problem. 7. Existence and uniqueness of a solution.

8. The energy of an equation. Variational principle.

9. Application: The deflection of simply supported and clamped beams.

10. Energy and stability.

11. Selected partial differential equations: Laplace, heat and wave equations.

12. Selected partial differential equations: Laplace, heat and wave equations.

Study Objective:

The primary goal is to get acquainted with mathematical modeling of phenomena in mechanics that are subject to other courses and provide basic math apparatus used in these courses. The topics include:

Matrix theory aimed towards methods of solutions of large systems of linear equations that occur in the course of numerical solutions of mechanics problems.

Boundary value problems for ordinary differential equations, namely those that model beam behavior.

Selected partial differential equations: Laplace, heat and wave equations.

Study materials:

[1] Rektorys, K.: Variational methods in mathematics, science and engineering. Translated from the Czech by Michael Basch. Second edition. D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1980.

[2] Rektorys, K.: Survey of applicable mathematics. Vol. II. Mathematics and its Applications, 281. Kluwer Academic Publishers Group, Dordrecht, 1994.

[3] Bubeník, F.: Mathematics for Engineers, textbook of Czech Technical University, Prague 2007

Note:

Further information:

https://mat.fsv.cvut.cz/zindulka/m4.html

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans: