Course Details

Applied Mathematics

Academic Year 2024/25

NAB023 course is part of 1 study plan

​ NPC-SIK Summer Semester 1st year

Mathematical approaches to the analysis of engineering applications, namely initial and boundary problems for ordinary and partial differenctial equations.

Credits

4 credits

Language of instruction

Czech

Semester

summer

Course Guarantor

prof. Ing. Jiří Vala, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

Forms and criteria of assessment

course-unit credit and examination

Entry Knowledge

Basics from the mathematical analysis, due to the courses for bachelor students at MAT FCE, and numerical methods.

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.

Basic Literature

DRÁBEK P., HOLUBOVÁ G.: Parciální diferenciální rovnice. ZČU v Plzni 2011 (cs)

Offered to foreign students

Not to offer

Course on BUT site

https://www.vut.cz/en/students/courses/detail/281902

Lecture

13 weeks, 2 hours/week, elective

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

13 weeks, 2 hours/week, compulsory

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.