Course detail

Mathematics II

FAST-DA02Acad. year: 2022/2023

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.

Language of instruction

Czech

Mode of study

Not applicable.

Guarantor

prof. Ing. Jiří Vala, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Not applicable.

Prerequisites

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

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

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.

Work placements

Not applicable.

Aims

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.

Specification of controlled education, way of implementation and compensation for absences

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

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Dalík Josef: Numerické metody. Akademické nakladatelství CERM, s.r.o. Brno 1997

Recommended reading

Čermák L.: Numerické metody II. Akademické nakladatelství CERM, s.r.o. Brno 2004
Marčuk G.I.: Metody numerické matematiky. Academia 1987
Míka S., Přikryl P.: Numerické metody řešení obyčejných diferenciálních rovnic. ZČU Plzeň 1994
Míka S., Přikryl P.: Numerické metody řešení parciálních diferenciálních rovnic. ZČU Plzeň 1995
Vitásek E.: Základy teorie numerických metod pro řešení diferenciálních rovnic. Academia Praha 1994
Ženíšek A.: Matematické základy metody konečných prvků. PC-DIR Brno 1997

Classification of course in study plans

  • Programme D-P-C-GK Doctoral

    branch GAK , 2 year of study, winter semester, compulsory-optional

  • Programme D-K-C-GK Doctoral

    branch GAK , 2 year of study, winter semester, compulsory-optional

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

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.