Course detail

Applied Mathematics

FAST-CA057Acad. year: 2022/2023

Basics of ordinary fifferential equations focussing on engineering applications – classic solution, Cauchy problem and boundary problems (their classification). Analytical methods for solving boudary problems in ordinary secod and fourth order differential equations.
Methods of solution of non-homogeneous boundary problems – Fourier method, Green´s function, variation of constants method. Solutions of non-linear differential equations with given boundary conditions. Sobolev spaces and generalized solutions and reason for using such notions. Variational methods of solutions.
Introduction to the theory of partial differential equations of two variables – classes and basic notions. Classic solution of a boundary problem (classes), properties of solutions.
Laplace and Fourier transform – basic properties.
Fourier method of solution of evolution equations, difussion problems, wave equation.
Laplace method used to solve evolution equations - heat transfer equation.
Equations used in the theory of elasticity.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

prof. Ing. Jiří Vala, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

The students manage the subject to the level of understanding foundation of the modern methods of ordinary and partial differential equations in the engineering applications.

Prerequisites

Basics of the theory of one- and more-functions. Differentiation and integration of functions.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

1. Basics of ordinary differential equations focussing on engineering applications – classic solution, Cauchy problem and boundary problems (classes).
2. Analytical methods used to solve boundary problems in ordinary secod and fourth order differential equations.
3. Methods of solution of non-homogeneous boundary problems – Fourier method,
4. Green´s function, variation of constants method.
5. Solutions of non-linear differential equations with given boundary conditions.
6. Sobolev spaces and generalized solutions and reason for using such notions.
7. Variational methods of solutions.
8. Introduction to the theory of partial differential equations of two variables – classes and basic notions.
9. Classic solution of a boundary problem (classes), properties of solutions.
10. Laplace and Fourier transform – basic properties.
11. Fourier method used to solve evolution equations, difussion problems, wave equation.
12. Laplace method used to solve evolution equations - heat transfer equation.
13. Equations used in the theory of elasticity.

Work placements

Not applicable.

Aims

Understanding the notion of generalized solutions to ordinary differential equations. Getting acquainted with principles of the modern methods used to solve odrinary and partial differential equations in transport structures.

Specification of controlled education, way of implementation and compensation for absences

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme N-P-C-SI Master's

    branch K , 1 year of study, summer semester, compulsory-optional

  • Programme N-K-C-SI Master's

    branch K , 1 year of study, summer semester, compulsory-optional

  • Programme N-P-E-SI Master's

    branch K , 1 year of study, summer semester, compulsory-optional

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

Syllabus

1. Basics of ordinary differential equations focussing on engineering applications – classic solution, Cauchy problem and boundary problems (classes). 2. Analytical methods used to solve boundary problems in ordinary secod and fourth order differential equations. 3. Methods of solution of non-homogeneous boundary problems – Fourier method, 4. Green´s function, variation of constants method. 5. Solutions of non-linear differential equations with given boundary conditions. 6. Sobolev spaces and generalized solutions and reason for using such notions. 7. Variational methods of solutions. 8. Introduction to the theory of partial differential equations of two variables – classes and basic notions. 9. Classic solution of a boundary problem (classes), properties of solutions. 10. Laplace and Fourier transform – basic properties. 11. Fourier method used to solve evolution equations, difussion problems, wave equation. 12. Laplace method used to solve evolution equations - heat transfer equation. 13. Equations used in the theory of elasticity.

Exercise

26 hod., compulsory

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

Syllabus

Related directly to the above listed topics of lectures. 1. Basics of ordinary differential equations focussing on engineering applications – classic solution, Cauchy problem and boundary problems (classes). 2. Analytical methods used to solve boundary problems in ordinary secod and fourth order differential equations. 3. Methods of solution of non-homogeneous boundary problems – Fourier method, 4. Green´s function, variation of constants method. 5. Solutions of non-linear differential equations with given boundary conditions. 6. Sobolev spaces and generalized solutions and reason for using such notions. 7. Variational methods of solutions. 8. Introduction to the theory of partial differential equations of two variables – classes and basic notions. 9. Classic solution of a boundary problem (classes), properties of solutions. 10. Laplace and Fourier transform – basic properties. 11. Fourier method used to solve evolution equations, difussion problems, wave equation. 12. Laplace method used to solve evolution equations - heat transfer equation. 13. Equations used in the theory of elasticity.