Course detail
Mathematical Analysis
FSI-UMA-A Acad. year: 2023/2024 Winter semester
The course provides an introduction to the theory of differential equations and dynamical systems. These branches form the theoretical background for mathematical modeling in physics, mechanics, and other disciplines.
Language of instruction
English
Number of ECTS credits
7
Supervisor
Department
Entry knowledge
Linear algebra, differential calculus of functions of one and several variables, integral calculus of functions of one variable, first order ordinary differential equations.
Rules for evaluation and completion of the course
Attendance at lectures and seminars is obligatory, attendance at seminars is checked. Absence from seminars may be compensated for by the agreement with the teacher.
Course-unit credit is awarded on the following conditions: A semestral project consisting of the assigned problems. Active participation in seminars.
Examination: The exam tests the knowledge of definitions and theorems (especially the ability of their application to the given problems) and practical skills in solving particular problems. The exam has written and oral part. For the written exam, one sheet of A4 hand-written paper (two-sided) is permitted with formulas and criteria of your choice (without particular examples). The use of a (simple) calculator is also allowed, but phones and computers are not permitted. The list of topics for the oral part of the exam will be announced at the end of semestr.
The final grade reflects the result of the examinational test (maximum 70 points), discussion about the examinational test (maximum 10 points), and the evaluation of the oral part (maximum 20 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).
Aims
The aim of the course is to acquaint the students with basic notions and methods of solving of ordinary differential equations, with the fundamentals of the theory of stability of solutions to autonomous systems, and with other selected topics from the theory of ordinary differential equations. The task is also to show that the knowledge of the theory of ordinary differential equations can frequently be utilised in physics, technical mechanics, and other branches.
Students will acquire skills for analytical solving of higher order ordinary differential equations and systems of first order ordinary differential equations. They will be able to examine the stability of the equilibria (singular points) of non-linear autonomous systems. Students will be also enlightened on ordinary differential equations as mathematical models and on the qualitative analysis of the obtained equations.
The study programmes with the given course
Programme N-ENG-A: Mechanical Engineering, Master's, compulsory
Type of course unit
Lecture
39 hours, optionally
Teacher / Lecturer
Syllabus
Higher-order ordinary differential equations (ODE). Basic notions. The existence and uniqueness of a solution to the initial value problem. General solutions of homogeneous and non-homogeneous linear equations.
Methods of solving of higher-order homogeneous linear ODEs with constant coefficients.
Solving of higher order non-homogeneous linear ODEs with constant coefficients – methods of undetermined coefficients and variation of parameters.
Systems of first-order ordinary differential equations. The xistence and uniqueness of a solution to the initial value problem. General solutions of homogeneous and non-homogeneous linear systems.
Methods of solving of homogeneous systems of linear ODEs with constant coefficients.
Solving of non-homogeneous systems of linear ODEs with constant coefficients – methods of undetermined coefficients and variation of parameters.
Stability of solutions to ordinary differential equations and their systems. Basic notions. Stability of linear systems of ODEs with constant coefficients.
Autonomous systems of first order ODEs. Trajectory and phase portrait. Equilibrium and its stability. Linearization.
Two-dimensional linear systems of ODEs with a constant regular matrix. Classification of equilibria.
Two-dimensional autonomous non-linear systems of ODEs. Topological equivalence.
Hamiltonian systems and second-order autonomous non-linear equations.
Mathematical modeling in mechanics and biology.
Exercise
26 hours, compulsory
Teacher / Lecturer
Syllabus
Analytical methods of solving of higher-order ODEs.
Analytical methods of solving of systems of first order ODEs.
Stability of linear systems of ODEs with constant coefficients.
Autonomous systems of first-order ODEs.
Two-dimensional linear systems of ODEs with a constant regular matrix – stability and classification of equilibria.
Two-dimensional autonomous non-linear systems of ODEs – stability and classification of equilibria.
Autonomous non-linear second-order equations – stability and classification of equilibria.
Mathematical modeling in mechanics and biology.