Course detail

Applied Mathematics

FAST-CA057Acad. year: 2024/2025

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)

Entry knowledge

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

Rules for evaluation and completion of the course

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

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.
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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

Not applicable.

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.