Obične diferencijalne jednadžbe i primjene

master's thesis

• Title (english): Ordinary differential equations and applications
• Author: Irena Janjatović
• Mentor: Marko Radulović (mentor)
• Committee member: Marko Radulović (predsjednik povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu promatrali smo kako diferencijalnim jednadžama i sustavima možemo modelirati razne probleme. Izveli smo nekoliko modela, pronašli njihova rješenja i analizirali stabilnost rješenja. U prvom poglavlju smo definirali obične diferencijalne jednadžbe i sustave te naveli teoreme o egzistenciji, jedinstvenosti i stabilnosti rješenja. U drugom poglavlju smo modelirali i analizirali nekoliko modela korištenjem običnih diferencijalnih jednadžbi, dok smo u trećem poglavlju isto napravili korištenjem sustava diferencijalnih jednadžbi. Modelirali smo probleme društvene, biološke i financijske prirode. Promatrali smo kako eutanazijom i sterilizacijom kontrolirati pse lutalice, kako kapital i radna snaga utječu na proizvodnju, vremensko zadržavanje lijeka u krvotoku te promjenu broja zaraženih i osjetljivih pojedinaca za gripu i koleru. Vidjeli smo kako diferencijalne jednadžbe imaju široku primjenu u stvarnom svijetu i da su jako koristan alat za modeliranje promjena varijabli koje su nam od interesa u određenom problemu.

Abstract (english)

In this thesis, we looked at how we can model various problems with differential equations and systems. We performed several models, found their solutions and analyzed the stability of the solutions. In the first chapter, we defined ordinary differential equations and systems, and stated theorems on the existence, uniqueness and stability of solutions. In the second chapter we modeled and analyzed several models using ordinary differential equations, while in the third chapter we did the same using a system of differential equations. We modeled problems of a social, biological and financial nature. We observed how to control stray dogs by euthanasia and sterilization, how capital and labor affect production, the time retention of the drug in the bloodstream, and the change in the number of infected and susceptible individuals for influenza and cholera. We have seen that differential equations have wide application in the real world and that they are a very useful tool for modeling the changes of variables that are of interest to us in a particular problem.

• Keywords: teorem o egzistenciji; teorem o jedinstvenosti; teorem o stabilnosti rješenja
• Keywords (english): existence theorem; uniqueness theorem; solution stability theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:356238
• Study programme: Title: Financial and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-01 13:35:04