Course Details

Mathematics II

Academic Year 2022/23

DA02 course is part of 3 study plans

stud.semestr.null 1st year

D-P-C-GK Winter Semester 1st year

D-K-C-GK Winter Semester 1st year

Numerical methods for the initial-value problem for one ordinary differential equation (ODE) of order one and for systems of ODE of order one, absolute stability, variational formulation of boundary-value problems for ODE and partial DE of order two, discretization of elliptic differential problems by the finite difference and the finite element methods, numerical methods for the non-stationary parabolic and hyperbolic differential problems, numerical solution of a nonlinear differential boundary-value problem.

Course Guarantor

prof. Ing. Jiří Vala, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

Objective

Students should be acquainted with the basics of the theory of numerical solution of ordinary differential equations and systems of such equations and second-order partial differential equations. They should learn how to use numeric methods to solve such equations.

Syllabus

1. Formulation of the initial-value problem for ordinary differential equations of degree 1, basic properties, existence and uniqueness of solutions.
2. Basic numerical methods for the initial-value problems and their absolute stability.
3. Introduction to the variational calculus, basic spaces of integrable functions.
4. Classical and variational formulations of elliptic problems for ordinary differential equations of degree 2, basic physical meanings.
5. Standard finite difference method for elliptic problems for ordinary differential equations (ODE) of degree 2 and its stable modifications.
6. Approximation of boundary-value problems for ODE of degree 2 by the finite element method.
7. Classical and variational formulation of elliptic problems for ODE od degree 4 and approximation by the finite element method.
8. Classical and variational formulation of elliptic problems for partial differential equations od degree 2.
9. Finite element method for elliptic problems for partial differential equations od degree 2.
10. Finite volume method.
11. Discretization of non-stationary problems for degree 2 differential equations by the method of lines.
12. Mathematical models of flow. Nonlinear problems and problems with dominating convection.
13. Numerical methods for the models of flow.

Prerequisites

Differential and integral calculus, numerical linear algebra, interpolation and approximation of functions, numerical differentiation and intgration

Language of instruction

Czech

Credits

no credit

Semester

summer

Forms and criteria of assessment

examination

Specification of controlled instruction, the form of instruction, and the form of compensation of the absences

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Offered to foreign students

Not to offer

Course on BUT site

https://www.vut.cz/en/students/courses/detail/255810

Lecture

13 weeks, 3 hours/week, elective

Syllabus

1. Formulation of the initial-value problem for ordinary differential equations of degree 1, basic properties, existence and uniqueness of solutions.
2. Basic numerical methods for the initial-value problems and their absolute stability.
3. Introduction to the variational calculus, basic spaces of integrable functions.
4. Classical and variational formulations of elliptic problems for ordinary differential equations of degree 2, basic physical meanings.
5. Standard finite difference method for elliptic problems for ordinary differential equations (ODE) of degree 2 and its stable modifications.
6. Approximation of boundary-value problems for ODE of degree 2 by the finite element method.
7. Classical and variational formulation of elliptic problems for ODE od degree 4 and approximation by the finite element method.
8. Classical and variational formulation of elliptic problems for partial differential equations od degree 2.
9. Finite element method for elliptic problems for partial differential equations od degree 2.
10. Finite volume method.
11. Discretization of non-stationary problems for degree 2 differential equations by the method of lines.
12. Mathematical models of flow. Nonlinear problems and problems with dominating convection.
13. Numerical methods for the models of flow.