Course detail

# Applied Mathematics

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.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

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.

Prerequisites

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

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations - lectures, seminars.

Assesment methods and criteria linked to learning outcomes

Successful completion of the scheduled tests and submission of solutions to problems assigned by the teacher for home work. Unless properly excused, students must attend all the workshops. The result of the semester examination is given by the sum of maximum of 70 points obtained for a written test and a maximum of 30 points from the seminar.

Language of instruction

Czech

Work placements

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.

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.

Classification of course in study plans

• Programme N-P-E-SI (N) Master's

branch K , 1. year of study, summer semester, 4 credits, compulsory-optional

• Programme N-K-C-SI (N) Master's

branch K , 1. year of study, summer semester, 4 credits, compulsory-optional

• Programme N-P-C-SI (N) Master's

branch K , 1. year of study, summer semester, 4 credits, compulsory-optional

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

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 hours, compulsory

Teacher / Lecturer

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.