2 semester

1.       

Title of discipline

Generalized functions

2.       

Course of study, specialty

1, 1-31 80 03 Mathematics

3.       

Semester of training (for each semester a separate table)

2

4.       

Number of credits (curriculum)

3

5.       

Name and surname lecturer

Antonevich Anatoly Borisovich

6.       

Objectives of the study of the discipline

Familiarization of students with the main ways of mathematical formalization of conflict situations in the economic and social spheres and the principles of their solution.

Training in various effective ways to resolve conflict situations, taking into account the interests of the conflicting parties.

Increase the overall level of mathematical culture and improve the skills of using mathematical methods to solve applied problems.

As a result of studying the discipline, the trainee must know:

  • Definition of the generalized function;
  • convergence of sequences of generalized functions;
  • definition of the derivative of a generalized function;
  • Is the product of a generalized function by a smooth function;
  • definition of the convolution of a generalized function;
  • basic examples of singular generalized functions;

be able to:

  • check the convergence of sequences of generalized functions;
  • find the derivatives of given generalized functions;
  • to build fundamental solutions for the simplest differential equations;

own:

  • basic concepts of the theory of generalized functions;
  • actions with generalized functions.

7.       

Prerequisites

Fundamentals of mathematical analysis

Fundamentals of functional analysis

Differential equations

8.       

Contents of the discipline

Theme 1. The space of basic functions. Convergence, operations of differentiation and multiplication by a smooth function. Topology of the space of basic functions.

Theme 2. The space of generalized functions.

Definition of a generalized function. Convergence in the space of generalized functions. Differentiation of generalized functions. Differentiation of discontinuous functions. Multiplication by smooth functions. The basic generalized functions xα.

Theme 3. Convolution of generalized functions.

Fundamental solutions for differential equations with constant coefficients.

Theme 4. Topological vector spaces. Locally convex topological vector spaces. Topology in spaces of basic and generalized functions.

Theme 5. The Fourier transform of generalized functions. The Schwartz space of moderately growing distributions. The Fourier transform. Construction of fundamental solutions using the Fourier transform.

Theme 6. Periodic generalized functions. Spaces of basic periodic functions. Periodic generalized functions. Fourier series.

9.       

Recommended Reading

Basic

  1. Vladimirov VS Generalized functions in mathematical physics / M. Nauka. 1976.- 280s.
  2. Shilov, F.E. Mathematical analysis. Second special course. M. Nauka / 1965.
  3. Antonevich AB, Radyno Ya.V. Functional analysis and integral equations; BSU. – Minsk: BSU, 2013. – 511 p.

10.   

Methods of teaching

Lectures, UIRS

11.   

Language of learning

Russian

12.   

Conditions (requirements), routine control

Test papers

 

13.   

Appraisal form

credit