1.       

Course title

Numerical methods

2.       

Course of Study, Speciality

3,

1-31 03 09 Computer Mathematics and System Analysis

3.       

Semester

6

4.       

Credits

1

5.       

Lecturer

Azarov Alexey Ivanovich
Candidate of Physical and Mathematical Sciences, Associate Professor

6.       

Course goal

– construction of mathematical models, determination of their role and significance;

– familiarity with the basic principles of developing computational methods for typical and new mathematical models;

– study and development of the theory and applications of computing methods, their computer implementations;

– analysis of the reliability of numerical results, their interpretation and implementation.

7.       

Prerequisites

– algebra and number theory;

– Geometry, mathematical analysis;

– modern systems of computer modeling

– differential equations;

– equations of mathematical physics;

– integral equations;

– Mechanics

8.       

Course Topics

Numerical methods for solving LAA systems. Norms of vectors and matrices. Estimation of the error in the solution of the LAA systems. Direct methods. The Gauss method.

Calculation of eigenvalues and eigenvectors of matrices. Properties of eigenvectors and eigenvalues of matrices. Converting similarity. The Danilevsky method.

Solution of nonlinear equations and systems. Separation of roots. Method of dichotomy. Multiple roots. The roots of polynomials. Simple iteration method. The convergence condition and the rate of convergence. Newton’s method.

Numerical solution of the Cauchy problem for ordinary differential equations. One-step methods. Euler’s method. Estimate of the rate of convergence. Runge-Kutta methods. Multistep methods.

Numerical solution of boundary value problems for ordinary differential equations. Difference method for solving a boundary-value problem for a second-order equation. Methods for constructing difference schemes. Integro-interpolation method. The concept of compact difference schemes. Galerkin’s method. Approximation and convergence.

Numerical solution of Fredholm integral equations. Basic approaches to solving integral equations. The Fourier method for the numerical solution of integral equations of convolution type.

Construction and investigation of difference schemes for problems of mathematical physics. Difference schemes for the heat equation. The canonical form and the stability condition for two-layer difference schemes. Stability, approximation and convergence. Difference schemes for the transport equation. Spectral criterion of stability. Difference schemes for elliptic equations. The maximum principle

9.       

Recommended Literature

1.      Бахвалов Н.С., Жидков Н.П., Кобельков Г.М. Численные методы. – М.: Наука, 1987.

2.      Крылов В.И., Бобков В.В., Монастырный П.И. Вычислительные методы высшей математики. В 2 т. – Минск: Выш. шк., 1972, 1975.

3.      Крылов В.И., Бобков В.В., Монастырный П.И. Начала теории вычислительных методов. Интерполирование и интегрирование. – Минск: Наука и техника, 1983.

4.      Мысовских И.П. Лекции по методам вычислений: учеб. пособие. – СПб.: Изд–во С.-Петерб. ун-та, 1998.

5.      Сборник задач по методам вычислений: учеб. пособие / Под ред. П.И. Монастырного, – Минск: Изд. центр БГУ, 2007.

6.      Методы вычислений. Интерполирование и интегрирование: курс лекций / М.В. Игнатенко. – Минск: БГУ, 2006.

10.   

Teaching Methods

Passive, active, interactive, verbal, visual, problematic

11.   

Teaching language

Russian

12.   

Requirements, current control

– report on laboratory work;

– mathematical dictation;

– colloquium.

13.   

Method of certification

Credit