magistrsko delo
Rok Rožič (Author), Valery Romanovski (Mentor), Brigita Ferčec (Co-mentor)

Abstract

V tej magistrski nalogi obravnavamo teorijo Darbouxjeve integrabilnosti. V prvem poglavju opišemo osnovne pojme komutativne algebre, ki so potrebni za našo študijo. V drugem poglavju obravnavamo teorijo Darbouxjeve integrabilnosti za dvodimenzionalne sisteme navadnih diferencialnih enačb. Opišemo obstoj Darbouxjevih prvih integralov in Darbouxjevih integrirajočih množiteljev v odvisnosti od števila invariantnih krivulj ter predstavimo izračun invariantnih krivulj in z uporabo le-teh konstrukcijo Darbouxjevih prvih integralov in integrirajočih množiteljev. Nadalje v tem poglavju s pomočjo opisane teorije Darbouxjeve integrabilnosti poiščemo Darbouxjev prvi integral in integrirajoči množitelj dveh dvodimenzionalnih sistemov četrte stopnje. V tretjem poglavju opišemo Darbouxjevo teorijo za n-dimenzionalne sisteme in v zadnjem poglavju navedemo odprto vprašanje za ravninska polinomska vektorska polja in nekatere uporabnosti Darbouxjeve teorije.

Keywords

ideali;prvi integral;vektorska polja;diferencialne enačbe;invariantne krivulje;invariantne površine;Darbouxjev prvi integral;Darbouxjev integrirajoči množitelj;magistrska dela;

Data

Language: Slovenian
Year of publishing:
Typology: 2.09 - Master's Thesis
Organization: UM FNM - Faculty of Natural Sciences and Mathematics
Publisher: [R. Rožič]
UDC: 517.9(043.2)
COBISS: 22190600 Link will open in a new window
Views: 1107
Downloads: 100
Average score: 0 (0 votes)
Metadata: JSON JSON-RDF JSON-LD TURTLE N-TRIPLES XML RDFA MICRODATA DC-XML DC-RDF RDF

Other data

Secondary language: English
Secondary title: Darboux integrability of polynomial systems of ordinary differential equations
Secondary abstract: In this master thesis we review some aspects of the Darboux integrability. In the first chapter we describe some basic notions of commutative algebra, which are needed for our study. In the second chapter we consider the theory of Darboux integrability for two-dimensional systems of ordinary differential equations. We describe the existence of Darboux first integrals and Darboux integrating factors depending on the number of invariant curves and then we present the calculation of invariant curves and using them the construction of Darboux first integrals and Darboux integrating factors. Then, using the described methods we find the Darboux first integral and the Darboux integrating factor for two systems inside of the family of two-dimensional systems of degree four. In the third chapter we describe the Darboux theory for n-dimensional systems and finally, in the last chapter we state an open question for planar polynomial vector fields and some applications of Darboux theory.
Secondary keywords: ideals;first integral;vector fields;differential equations;invariant curves;invariant hypersurfaces;Darboux first integral;Darboux integrating factor;master theses;
URN: URN:SI:UM:
Type (COBISS): Master's thesis/paper
Thesis comment: Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages: IX, 47 f.
ID: 9133623