Course details

Selected parts from mathematics II.

XPC-VPM FEKT XPC-VPM Acad. year 2024/2025 Summer semester 5 credits

Current academic year

The aim of this course is to introduce the basics of calculation of improper multiple integral and basics of solving of linear differential equations using delta function and weighted function.
In the field of improper multiple integral, main attention is paid to calculations of improper multiple integrals on unbounded regions and from unbounded functions.
In the field of linear differential equations, the following topics are covered: Eliminative solution method, method of eigenvalues and eigenvectors, method of variation of constants, method of undetermined coefficients, stability of solutions.

Guarantor

Šmarda Zdeněk, doc. RNDr., CSc. (UMAT)

Course coordinator

Kovár Martin, doc. RNDr., Ph.D. (UMAT)

Language of instruction

Czech

Completion

Examination

Time span

  • 26 hrs lectures
  • 26 hrs exercises

Department

Department of Mathematics (UMAT)

Learning objectives

The aim of this course is to introduce the basics of improper multiple integrals, systems of differential equations including of investigations of a stability of solutions of differential equations and applications of selected functions with solving of dynamical systems.
Students completing this course should be able to:
- calculate improper multiple integral on unbounded regions and from unbounded functions.
- apply a weighted function and a delta function to solving of linear differential equations.
- select an optimal solution method for given differential equation.
- investigate a stability of solutions of systems of differential equations.

Prerequisite knowledge and skills

The student should be able to apply the basic knowledge of analytic geometry and mathamatical analysis on the secondary school level: to explain the notions of general, parametric equations of lines and surfaces and elementary functions.
From the BMA1 and BMA2 courses, the basic knowledge of differential and integral calculus and solution methods of linear differential equations with constant coefficients is demanded. Especially, the student should be able to calculate derivative (including partial derivatives) and integral of elementary functions.

Syllabus of lectures

  1. Impulse functions,  solving differential  equations using a weight function
  2. Systems  of  differential  equations,  Elimination  method
  3. Constants variation  method ,  Method  of eigenvalue numbers  and eigenvalue vectors
  4. Method of undetermined coefficients
  5. Differential  transform method (DTM) 
  6.  DTM  for systems of differential  equations, delayed systems
  7. Difference equations, rules for differences, summation
  8. Solving linear homogeneous and non-homogeneous difference equations
  9. Gamma function,  solving specific nonlinear  difference equations 
  10. Solving systems of difference equations
  11. Fractional  calculus,  Mittag-Leffler  functions
  12. Solving fractional differential  equations in the sense of  Caputo  a Riemann-Liouville  derivative
  13. Solving fractional systems of differential  equations,  impulse  characterizations

 

Syllabus of numerical exercises

  1. Impulse functions,  solving differential  equations using a weight function
  2. Systems  of  differential  equations,  Elimination  method
  3. Constants variation  method ,  Method  of eigenvalue numbers  and eigenvalue vectors
  4. Method of undetermined coefficients
  5. Differential  transform method (DTM) 
  6.  DTM  for systems of differential  equations, delayed systems
  7. Difference equations, rules for differences, summation
  8. Solving linear homogeneous and non-homogeneous difference equations
  9. Gamma function,  solving specific nonlinear  difference equations 
  10. Solving systems of difference equations
  11. Fractional  calculus,  Mittag-Leffler  functions
  12. Solving fractional differential  equations in the sense of  Caputo  a Riemann-Liouville  derivative
  13. Solving fractional systems of differential  equations,  impulse  characterizations

 

Progress assessment

The student's work during the semestr (written tests and homework) is assessed by maximum 30 points.
Written examination is evaluated by maximum 70 points. It consist of seven tasks (one from improper multiple integral (10 points), three from application of a weighted function and a delta function (3 X 10 points) and three from analytical solution method of differential equations (3 x 10 points)).
The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year.

Course inclusion in study plans

  • Programme BIT, 2nd year of study, Elective
  • Programme BIT (in English), 2nd year of study, Elective
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