Darbouxjeva integrabilnost polinomskih sistemov navadnih diferencialnih enačb

Rožič, Rok

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Title:	Darbouxjeva integrabilnost polinomskih sistemov navadnih diferencialnih enačb
Authors:	ID Rožič, Rok (Author) ID Romanovskij, Valerij (Mentor) More about this mentor... ID Ferčec, Brigita (Comentor)
Files:	MAG_Rozic_Rok_2016.pdf (560,46 KB) MD5: 9AEB1999CE2E343A19975D31489B276E
Language:	Slovenian
Work type:	Master's thesis/paper
Typology:	2.09 - Master's Thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
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:	Ideal, prvi integral, vektorsko polje, diferencialna enačba, invariantna krivulja, invariantna površina, Darbouxjev prvi integral, Darbouxjev integrirajoči množitelj.
Place of publishing:	Maribor
Publisher:	[R. Rožič]
Year of publishing:	2016
PID:	20.500.12556/DKUM-58037
UDC:	517.9(043.2)
COBISS.SI-ID:	22190600
NUK URN:	URN:SI:UM:DK:F5PNTGUB
Publication date in DKUM:	12.05.2016
Views:	2046
Downloads:	159
Metadata:
Categories:	FNM
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Secondary language

Language:	English
Title:	Darboux integrability of polynomial systems of ordinary differential equations
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.
Keywords:	Ideal, first integral, vector field, differential equation invariant curve, invariant hypersurfaces, Darboux first integral, Darboux integrating factor.

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