Course detail
Mathematics III
FSI-3MAcad. year: 2022/2023
The course provides an introduction to the theory of infinite series and the theory of ordinary and partial differential equations. These branches form the theoretical background in the study of many physical and engineering problems. The course deals with the following topics:
Number series. Function series. Power series. Taylor series. Fourier series.
Ordinary differential equations. First order differential equations. Higher order linear differential equations. Systems of first order linear differential equations.
Partial differential equations. Classification. Modelling with differential equations.
Basic numerical methods for solving differential equations with a suitable software (e.g. Matlab).
Language of instruction
Number of ECTS credits
Mode of study
Guarantor
Department
Learning outcomes of the course unit
Prerequisites
Co-requisites
Planned learning activities and teaching methods
Assesment methods and criteria linked to learning outcomes
Examination: The examination tests the knowledge of definitions and theorems (especially the ability of their application to the given problems) and practical skills in solving of examples. The exam is written (possibly followed by an oral part). The written exam consists in particular of the examples on the following topics:
Number and function series, the expansion of a function into Taylor series, solving of first order ODEs, solving of higher order linear ODEs, solving of system of first order linear ODEs, Fourier series, solving of ODEs via the infinite series and the Laplace tranform method, boundary value problems, basics of PDEs theory. Some theoretical queries concerning basic concepts can be included in the written part as well.
Topics of practical part: The expansion of a function into Taylor series, solving of first order ODEs, solving of higher order linear ODEs, solving of system of first order linear ODEs.
The final grade reflects the result of the written part of the exam (maximum 65 points), the result of the oral part (maximum 15 points), the results achieved in seminars (maximum 14 points), and the results achieved in seminars in computer labs (maximum 6 points).
Grading scheme is as follows: excellent (90-100 points), very good
(80-89 points), good (70-79 points), satisfactory (60-69 points), sufficient (50-59 points), failed (0-49 points).
Course curriculum
Work placements
Aims
Specification of controlled education, way of implementation and compensation for absences
Recommended optional programme components
Prerequisites and corequisites
Basic literature
Hartman, P.: Ordinary Differential Equations, New York, 1964. (EN)
Kalas, J., Ráb, M: Obyčejné diferenciální rovnice, PřF MU, Brno, 2012. (CS)
Recommended reading
Čermák, J., Nechvátal, L.: Matematika III, Brno, 2016. (CS)
Čermák, J.: Sbírka příkladů z Matematické analýzy III a IV, Brno, 1998. (CS)
Kalas, J., Ráb, M: Obyčejné diferenciální rovnice, PřF MU, Brno, 2012. (CS)
Logan, J.D.: A First Course in Differential Equations. New York, Springer, 2006. (EN)
Elearning
Classification of course in study plans
- Programme B-MET-P Bachelor's 2 year of study, winter semester, compulsory
- Programme B-ZSI-P Bachelor's
specialization STI , 2 year of study, winter semester, compulsory
specialization MTI , 2 year of study, winter semester, compulsory - Programme LLE Lifelong learning
branch CZV , 1 year of study, winter semester, compulsory
Type of course unit
Lecture
Teacher / Lecturer
Syllabus
2. Operations with number series. Function series. Basic properties.
3. Power series. Taylor series and expansions of functions into power series.
4. Trigonometric Fourier series. Problems of the convergence and expansions of functions.
5. Ordinary differential equations (ODE). Basic notions. The existence and uniqueness of the solution to the initial value problem for 1st order ODE. Analytical methods of solving of 1st order ODE.
6. Higher order ODEs. Basic notions. The existence and uniqueness of the solution to the initial value problem for higher order ODEs. Genral solutions of homogeneous and nonhomogeneous linear equations. Methods of solving of higher order homogeneous linear ODEs with constant coefficients.
7. Methods of solving of higher order non-homogeneous linear ODEs with constant coefficients.
8. Systems of 1st order ODEs. Basic notions. The existence and uniqueness of the solution to the initial value problem for systems of 1st order ODE. General solution of homogeneous and non-homogeneous systems of 1st order ODE.
9. Methods of solving of homogeneous systems of 1st order linear ODEs.
10. Methods of solving of non-homogeneous systems of 1st order linear ODEs.
11. The Laplace transform and its use in solving of linear ODEs. The method of Taylor series in solving of ODEs.
12. Stability of solutions of ODEs and their systems. Boundary value problem for 2nd order ODEs. Partial differential equations. Basic notions. The equations of mathematical physics.
13. Mathematical modelling by differential equations.
Exercise
Teacher / Lecturer
Syllabus
2. Infinite series.
3. Function and power series.
4. Taylor series.
5. Fourier series.
6. Analytical methods of solving of 1st order ODEs.
7. Analytical methods of solving of 1st order ODEs (continuation).
8. Higher order linear homogeneous ODEs.
9. Higher order non-homogeneous linear ODEs.
10. Systems of 1st order linear homogeneous ODEs.
11. Systems of 1st order linear non-homogeneous ODEs.
12. Systems of 1st order linear ODEs - continuation.
13. Laplace transform method and series method of solving of ODEs.
Computer-assisted exercise
Teacher / Lecturer
Syllabus
Elearning