Course detail

Selected parts from mathematics II.

FEKT-XPC-VPMAcad. year: 2025/2026

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.

Language of instruction

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

doc. RNDr. Zdeněk Šmarda, CSc.

Department

Department of Mathematics (UMAT)

Entry knowledge

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.

Rules for evaluation and completion of the course

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.

Aims

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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

HLAVIČKOVÁ, I. KOLÁŘOVÁ, E., ŠMARDA,Z., Vybrané partie z matematiky II. -učební text (CS)

Type of course unit

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Zdeněk Šmarda, CSc.

Syllabus

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

Fundamentals seminar

12 hod., compulsory

Teacher / Lecturer

doc. RNDr. Zdeněk Šmarda, CSc.

Syllabus

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

Computer-assisted exercise

14 hod., compulsory

Teacher / Lecturer

doc. RNDr. Zdeněk Šmarda, CSc.

Syllabus

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

VUT

Faculties

University Institutes

Parts

Selected parts from mathematics II.

Type of course unit