Numeričke metode za rješavanje običnih diferencijalnih jednadžbi

master's thesis

• Title (english): Numerical methods for solving ordinary differential equations
• Author: Ana Grbeš
• Mentor: Tina Bosner (mentor)
• Committee member: Tina Bosner (predsjednik povjerenstva)
• Committee member: Goran Igaly (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Committee member: Goranka Nogo (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-09-21, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu proučavali smo jednokoračne i višekoračne metode za rješavanje inicijalnog problema za obične diferencijalne jednadžbe. Ovakve metode su u 20. stoljeću jako napredovale u numeričkoj analizi posebno zbog kompjuterizacije i potrebe za matematičkim modeliranjem efektivnih numeričkih algoritama. Definirali smo jednokoračne metode i analizirali konvergenciju tih metoda uz pomoć konzistencije i stabilnosti. Od jednokoračnih metoda spomenuli smo Eulerovu metodu, poboljšanu Eulerovu, midpoint te Runge-Kutta metode. Za svaku metodu na konkretnom primjeru računali smo aproksimacije te greške metoda, te smo na kraju dali usporedbu grešaka tih metoda. Za višekoračne metode dana je definicija te je analizirana stabilnost i konvergencija. Od višekoračnih metoda spomenuli smo Adamsove, BDF metode, prediktor-korektor metode i krute sustave. Za Adams- Bashforth-ove i prediktor-korektor metode na konkretnom primjeru računali smo aproksimacije te greške metoda.

Abstract (english)

In this paper we are examining one-step and multistep numerical methods for solving initial value problem for ordinary differential equations. These methods made enormous progress during the 20th century in numerical analysis especially because of electronic computers and the needs in mathematical modelling of efficient numerical algorithms. We defined one-step methods and analyzed the convergence of these methods with the help of consistency and stability. As part of one-step methods, we mentioned the Euler method, the improved Euler method, the midpoint, and Runge-Kutta methods. For each method on a specific example, we calculated the approximations and errors of the methods, and finally gave a comparison of the errors of these methods. As part of multi-step methods, a definition is given and stability and convergence are analyzed. Of the multi-step methods, we mentioned Adams, BDF methods, predictor-corrector methods, and stiff systems. For the Adams-Bashforth and predictor-corrector methods, we calculated approximations of this method error on a concrete example.

• Keywords: Eulerova metoda; midpoint metoda; Runge-Kutta metoda; višekoračne metode; jednokoračne metode; Adamsove metode; BDF metode; prediktor-korektor metode
• Keywords (english): Euler method; improved Euler method; midpoint method; Runge-Kutta method; Adams method; BDF method; predictor-corrector method; multi-step methods; one-step methods
• Language: croatian
• URN:NBN: urn:nbn:hr:217:597808
• Study programme: Title: Mathematics and Computer Science Education; specializations in: Mathematics and Computer Science Education; Course: Mathematics and Computer Science Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike i informatike (magistar/magistra edukacije matematike i informatike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-10-27 10:00:48