Perturbacije diferencijalnih jednadžbi

master's thesis

• Author: Ivan Vidović
• Mentor: Eduard Marušić-Paloka (mentor)
• Committee member: Eduard Marušić-Paloka (predsjednik povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Committee member: Ana Prlić (član povjerenstva)
• Committee member: Snježana Lubura Strunjak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-07-15, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Na početku rada, u poglavlju Regularne perturbacije, pokazana je osnovna ideja perturbacijske metode ilustrirana jednostavnom algebarskom jednadžbom čije je egzaktno rješenje unaprijed poznato. Promatranjem fizikalnog procesa, u primjeru Oscilator, i korištenjem regularnih perturbacija došli smo do nezadovoljavajućeg aproksimativnog rješenja zbog pojave sekularnih članova. Poincaré-Lindstedtovom metodom korigirali smo singularno ponašanje koje se pojavljuje u svim slučajevima periodičkog gibanja. U asimptotičkoj analizi definirali smo pojmove vezane uz konvergenciju i uniformnost kako bismo nadopunili analizu aproksimativnog rješenja. U drugom poglavlju, bavili smo se metodom singularnih perturbacija. Prvo smo uveli pojam dominantnog uravnoteženja kojim smo objasnili izostanak rješenja kod regularnih perturbacija. Zatim smo, kroz primjere, proučavali graf egzaktnog rješenja rubnog problema i predstavili pojmove: rubni sloj, vanjski sloj te unutarnja i vanjska aproksimacija. U posljednjem, trećem poglavlju, proveli smo analizu rubnog sloja te uskladili vanjsku i unutarnju aproksimaciju pomoću uvjeta usklađivanja. Na samom kraju rada dan je opći postupak za rješavanje rubnih problema.

Abstract (english)

In the chapter entitled Regular perturbations at the beginning of this thesis, the basic idea of the perturbation method, illustrated by a simple algebraic equation, in which the exact solution is known upfront, has been demonstrated. By observing the physical process during the Oscillator example and using regular perturbations we reached an approximate solution which was unsatisfactory due to the occurrence of secular terms. The singular behaviour which appears in all periodic motion has been corrected by using Poincaré-Lindstedt method. In order to complement the analysis of the approximate solution, terms related to convergence and uniformity have been defined in the asymptotic analysis. In the second chapter, the singular perturbation method has been dealt with. Firstly, the term dominant balancing was introduced, allowing for the explanation of the absence of the solution in the regular perturbations. Then we studied, across examples, the graph of the exact solution to the boundary problem and introduced the following terms: boundary layer, outer layer and inner and outer approximation. In the third and last chapter we conducted an analysis of the boundary layer and brought into conformity the outer and inner approximation through matching conditions. At the very end of the thesis a general procedure for solving boundary problems has been provided.

• Keywords: perturbacijska metoda; regularne perturbacije; Poincaré-Lindstedtova metoda; rubni problemi
• Keywords (english): perturbation method; regular perturbations; Poincaré-Lindstedt method; boundary problem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:832642
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-09-18 12:59:14