Differential equations

Garant předmětu:

Přednášející:

Cvičící:

Předmět zajišťuje:

katedra aplikované matematiky

Anotace:

This course provides a foundational overview of differential equations, starting with basic motivation and examples of ODEs and progressing to essential solution methods like separation of variables. Key theorems on existence and uniqueness establish when solutions can be guaranteed. Linear and system-based ODEs are covered with methods like characteristic polynomial analysis, followed by examples of non-linear models such as predator-prey and epidemiological models to showcase real-world applications. Finally, an introduction to partial differential equations (PDEs) extends these concepts to multi-variable contexts.

The course will also cover numerical methods for solving ODEs and PDEs, including implicit and explicit Euler methods, Runge-Kutta methods, and finite element methods for both ODEs and PDEs.

Požadavky:

It is recommended to be comfortable with topics covered by BIE-LA1, BIE-MA1, and BIE-MA2 courses.

Osnova přednášek:

1 Motivation and first examples of ordinary differential equations (ODEs)

2 Separation of variables

3 The Cauchy problem, theorems on the existence and uniqueness of solution

4 Linear ODEs

5 Systems of linear ODEs

6 Explicit and implicit Euler methods, stability domains

7 Runge-Kutta methods, applications to systems of ODEs

8 Examples of non-linear models (predator-prey model, epidemiological models)

9 Introduction to partial differential equations (PDEs)

10 Boundary value problems

11 Shooting method, finite differences, finite element method for ODEs

12 Finite element method for PDEs

Osnova cvičení:

1 Motivation and first examples of ordinary differential equations (ODEs)

2 Separation of variables

3 The Cauchy problem, theorems on the existence and uniqueness of solution

4 Linear ODEs

5 Systems of linear ODEs

6 Explicit and implicit Euler methods, stability domains

7 Runge-Kutta methods, applications to systems of ODEs

8 Examples of non-linear models (predator-prey model, epidemiological models)

9 Introduction to partial differential equations (PDEs)

10 Boundary value problems

11 Shooting method, finite differences, finite element method for ODEs

12 Finite element method for PDEs

Cíle studia:

Studijní materiály:

1. D. Schaeffer and J. Cain, Ordinary Differential Equations: Basics and Beyond, Springer-Verlag New York Inc., 2016

2. Braun M., Differential equations and their applications: An Introduction to Applied Mathematics, Spinger, 1992

3. L. C. Evans: Partial Differential Equations, 2nd ed., American Mathematical Society, Rhode Island, 2010.

Poznámka:

Information about the course and courseware are available at https://courses.fit.cvut.cz/BIE-DIF/

Další informace:

https://bk.fit.cvut.cz/cz/predmety/00/00/00/00/00/00/08/09/91/p8099106.html

Pro tento předmět se rozvrh nepřipravuje

Předmět je součástí následujících studijních plánů: