Numerical Methods

2024/2025

Academic Year

ENG

Instruction in English

3

ECTS credits

Delivered at:

School of Applied Mathematics

Course type:

Compulsory course

When:

3 year, 3, 4 module

Instructors

Брандышев Петр Евгеньевич

Rakhel, Mark

Course Syllabus

Abstract

In modern natural sciences the numerical methods play a crucial role. It is related to the fact that modern problems arising in fundamental researches as well as in different applications as a rule have a complicated nature. For example, in such areas as Molecular Biology or Physical Chemistry usually researchers deal with the complex systems containing huge numbers of interacting each other molecules. The problems of such kind results in a serious motivation to develop the reliable numerical methods that are allow solving different problems, which may be reduced to the well-posed mathematical tasks. In this respect, the numerical methods in modern education of mathematicians become indispensable.The present course contains two parts. The first part (3rd year, 3-4 modules) contains seven large independent topics. Within the first topic will discuss to the methods of numerical solving of nonlinear equations, such as a method of simple iterations, tangent method, bisection method, and golden section method. In particular, to be discussed the method of localizing roots. The second topic is related to the matrix and iterative methods for solving of systems of linear equations, such as Gaussian method, Sweep method, method of Simple Iteration, Gauss-Seidel method, Iterative method with optimal parameter. The convergence theorems of iterative methods that are mentioned above will be discussed. Within the third topic will be discussed the interpolation methods such as Least square and Polynomial interpolation. In particular, the interpolation of functions by means of the Lagrange polynomials and Splines shall be considered. The fourth topic is about the methods of numerical solving of system of nonlinear equations. Within this topic shall be discussed in particular the Newton’s method and method of Simple iterations. Moreover, the theorem of convergence of these iterative schemes also shall be considered. The fifth topic is about the method of minimization of the functions of a one variable. The Bisection method and method of Golden section shall be discussed. The sixth topic is related to the numerical integration and differentiation of the functions of a one variable. In particular, the rectangle method, trapeze method, and Simpson method of numerical integration shall be studied. In addition, the finite difference formulas for first and second derivatives of the functions of a one variable shall be discussed.The second part (4th year, 1-2 modules) contains three large independent topics. The first topic is about the numerical solving of the Cauchy problem. Within this topic shall be discussed the following subjects: (1) the statement of Cauchy problem; (2) geometrical sense of the Cauchy problem; (3) Euler method; (4) Picard method; (5) Method of expansion in Taylor series; Runge-Kutta-Method. The second topic is related to the numerical calculation of the Boundary value problem. Within the second topic shall be discussed the following subjects: (1) The statement of the Boundary value problem; (2) The choosing of the coordinate grid; (3) the building of finite-difference scheme; (4) Stability and convergence of the finite-difference scheme; (5) The shooting method. The third topic is related to the numerical solving of the partial differential equations such as one-dimensional heat equation and wave equation. Within this topic shall be discussed the following subjects: (1) The statement of the initial-boundary value problem for one-dimensional heat equation; (2) The statement of the initial-boundary value problem for one-dimensional heat equation; (3) The explicit and implicit finite-difference schemes for the solving of one-dimensional heat equation; (4) The explicit and implicit finite-difference schemes for the solving of one-dimensional wave equation; (5) Numerical realization of the explicit and implicit finite-difference schemes.