Numerical integration and ordinary differential equations

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M4
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Numerical differentiation: Stable computing of derivatives. Differential approximations for derivatives.
Numerical integration: Degree of a rule and convergence. Newton-Cotes integration rules. Gauss quadratures. Composite rules. Error estimates. Convergence. Euler-Maclaurin formula and Romberg extrapolation. Singular integrals. Multiple integrals. Monte Carlo methods.
Ordinary differential equations:
Initial value problems. First order ODE equations. Higher order ODE equations. Systems of ODE equations. Local and global error. Explicit and implicit methods.
One-step methods: Euler method. Methods based on Taylor's series. Runge-Kutta methods. Explicit RK method of order four, trapezoidal rule, Merson method, Runge-Kutta Fehlberg method. Stability, consistency and convergence of one-step methods. A-stability.
Multistep methods: Methods based on numerical integration. Adams methods. Linear multistep methods. Characteristic polynomials and a local error. Predictor-Corrector methods. Milne's method. Zero stability. Order estimation of a zero stable method. Methods based on derivative approximations. Implicit BDF methods. Stability, consistency and convergence of multistep methods.
Introduction to Lie group methods.
Boundary value problems: Linear equations. Initial value and shooting methods. Finite difference methods.
Introduction to integral equations.

Readings

J. Kozak: Numerična analiza, DMFA-založništvo, Ljubljana, 2008.
R. L. Burden, J. D. Faires: Numerical Analysis, 8th edition, Brooks/Cole, Pacific Grove, 2005.
E. K. Blum: Numerical Analysis and Computation : Theory and Practice, Addison-Wesley, Reading, 1998.
Z. Bohte: Numerične metode, DMFA-založništvo, Ljubljana, 1991.
S. D. Conte, C. de Boor: Elementary Numerical Analysis : An Algorithmic Approach, 3rd edition, McGraw-Hill, Auckland, 1986.
E. Isaacson, H. B. Keller: Analysis of Numerical Methods, John Wiley & Sons, New York-London-Sydney, 1994.
D. R. Kincaid, E. W. Cheney: Numerical Analysis : Mathematics of Scientific Computing, 3rd edition, Brooks/Cole, Pacific Grove, 2002.

Objectives and competences

Student supplements knowledge of numerical differentiation, integration and numerical solving of ODE equations. By solving homeworks the obtained theoretical knowledge is consolidated.

Intended learning outcomes

Knowledge and understanding: Understanding methods for numerical integration and ordinary differential equations. Ability of numerical solving of initial and boundary value problems with the help of computers. Capability of choosing the most appropriate algorithm according to some features of the problem.
Application: Numerical computing of integrals ans numerical solving of ODE equations using a computer and error estimation based on theory. Problems that can not be solved any other way that numerically occurs very often
in practise (physics, mechanics, chemistry, economy...).
Reflection: Understanding of theory through applications.
Transferable skills: Skill of using computer for solving numerical problems. Understanding differences between exact and numerical computing. Knowledge of analysis and other fields of mathematics is constructively upgraded.

Learning and teaching methods

Lectures, exercises, homeworks, consultations.

Assessment

Type (homeworks, examination, oral, coursework, project):
homeworks or project
written exam
oral exam
Grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Marjetka Krajnc:
JAKLIČ, Gašper, KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito, ŽAGAR, Emil. High order parametric polynomial approximation of conic sections. Constructive approximation, ISSN 0176-4276, 2013, vol. 38, iss. 1, str. 1-18. [COBISS-SI-ID 16716121]
KRAJNC, Marjetka. Interpolation scheme for planar cubic G [sup] 2 spline curves. Acta applicandae mathematicae, ISSN 0167-8019, 2011, vol. 113, no. 2, str. 129-143. [COBISS-SI-ID 16215385]
KRAJNC, Marjetka. Geometric Hermite interpolation by cubic G[sup]1 splines. Nonlinear Analysis, Theory, Methods and Applications, ISSN 0362-546X. [Print ed.], 2009, vol. 70, iss. 7, str. 2614-2626. [COBISS-SI-ID 15508569]
Emil Žagar:
JAKLIČ, Gašper, KOZAK, Jernej, VITRIH, Vito, ŽAGAR, Emil. Lagrange geometric interpolation by rational spatial cubic Bézier curves. Computer Aided Geometric Design, ISSN 0167-8396, 2012, vol. 29, iss. 3-4, str. 175-188. [COBISS-SI-ID 16207449]
KOZAK, Jernej, ŽAGAR, Emil. On geometric interpolation by polynomial curves. SIAM journal on numerical analysis, ISSN 0036-1429, 2004, vol. 42, no. 3, str. 953-967. [COBISS-SI-ID 13398617]
ŽAGAR, Emil. On G [sup] 2 continuous spline interpolation of curves in R [sup] d. BIT, ISSN 0006-3835, 2002, vol. 42, no. 3, str. 670-688. [COBISS-SI-ID 12027993]