Description:

This course introduces students to the classical theory of ordinary differential equations (separable and linear ODEs) and also to bsics of numerical methods (errors in calculations and stability, numerical solutions of algebraic and differential equations and their systems). The course takes advantage of the synnergy between theoretical and practical point of view.

Contents:

1. Solving ODEs by separation. Slope field, stability of equilibria.
2. Errors in computing.
3. Approximating derivative, order of method.
4. Numerical integration.
5. Numerical solution of differential equations (Euler method, Runge-Kutta).
6. Linear ODEs - homogeneous and non-homogeneous (method of undetermined coefficients, variation method).
7. Numerical solution of higher order ODEs.
8. Numerical methods for finding roots of functions (bisection method, Newton method, iteration method).
9. Finite methods of solving systems of linear equations (GEM, LU decomposition). Complexity of algorithm. Stability.
10. Iteration methods for solving systems of linear equations (Gauss-Seidel).
11. Systems of ODEs. Stability of solutions.
12. Numerical methods for determining eigenvalues and eigenvectors of matrices.
13. Applications of differential equations.

Seminar contents:

1. Ordinary differential equations solvable by separation.
2. Analysis of solutions (stability, existence).
3. Getting to know the system, error in calculations.
4. Numerical integration.
5. Numerical solution of differential equations.
6. Homogeneous linear differential equations.
7. Equations with quasipolynomial right hand-side. Method of undetermined coefficients.
8. Numerical methods for finding roots of functions.
9. Homogeneous systems of linear ODEs.
10. Systems of linear ODEs.
11. Systems of linear ODEs numerically.
12. Eigenvalues and eigenvectors of matrices numerically.
13. Review of differential equations.

Recommended literature:

[1] Habala P.: Ordinary Differential Equations And Numerical Analysis, online, 2020, popřípadě kratší verze v češtině.
[2] Epperson, J.F.: An Introduction to Numerical Methods and Analysis. John Wiley & Sons, 2013.
[3] William E. Boyce, Richard C. DiPrima, Douglas B. Meade: Boyce's Elementary Differential Equations and Boundary Value Problems, 11. vydání, 2017.

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar