Fredholmova alternativa

master's thesis

• Author: Ante Sikirić
• Mentor: Eduard Marušić-Paloka (mentor)
• Committee member: Eduard Marušić-Paloka (predsjednik povjerenstva)
• Committee member: Dalibor Paar (član povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Committee member: Petar Žugec (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovome radu bavio sam se Fredholmovom alternativom i njezinim posljedicama. Fredholmova alternativa proučava Fredholmovu integralnu jednadžbu zajedno sa pridruženom transponiranom jednadžbom, te pridruženom homogenom i homogenom transponiranom jednadžbom. Uvjeti postojanja rješenja alternativom su podijeljeni u dva dijela ovisno o ispunjenju određenih uvjeta koje sama alternativa kao teorem nameće. Integralne jednadžbe s degeneriranim jezgrama, tj. jezgrama koje se mogu prikazati kao suma produkata funkcija samo varijable \(x\) i samo varijable \(t\) vode na sustav linearnih jednadžbi koji se onda proučavaju standardnim metodama, tj. koristi se teorem o rangu i defektu operatora. Slučaj sa općenitijim jezgrama, tj. ne nužno degeneriranim može se svesti na slučaj s degeneriranim jezgrama ako proizvoiljnu nedegeneriranu jezgru aproksimiramo degeneriranom. Metodama moderne funkcionalne analize Fredholmove integralne jednadžbe moguće je promatrati pomoću integralnih operatora te se navode svojstva takvih operatora. Uvođenjem integralnih operatora problem nalaženja rješenja integralne jednadžbe vodi na dobro poznati problem nalaženja svojstvenih vrijednosti i svojstvenih vektora nekog operatora.

Abstract (english)

In this paper I studied Fredholm alternative and its consequences. The Fredholm alternative studies the Fredholm integral equation together with the associated transposed equation, and the associated homogeneous and homogeneous transposed equation. The conditions for the existence of a solution by an alternative are divided into two parts, depending on the fulfilment of certain conditions imposed by the alternative itself as a theorem. Integral equations with degenerate kernels, i.e. kernels that can be represented as the sum products of functions of only \(x\) and only \(t\) variables lead to a system of linear equations, which are then studied by standard methods, i.e. the rank and defect theorem is used. A case with more general kernel, i.e. not necessarily degenerate, can be reduced to a case with degenerate kernel if we approximate the arbitrary non-degenerate kernel with a degenerate one. Using methods of modern functional analysis it is possible to study Fredholm integral equations with integral operators and we list some properties of these operators. With the introduction of integral operators, the problem of finding the solution of the integral equation leads to the well-known problem of finding the eigenvalues and eigenvectors of an operator.

• Keywords: Fredholmova alternativa; funkcionalna analiza; integralne jednadžbe; integralni operatori
• Keywords (english): Fredholm alternative; functional analysis; integral equations; integral operators
• Language: croatian
• URN:NBN: urn:nbn:hr:217:580888
• Study programme: Title: Mathematics and Physics Education; specializations in: Mathematics and Physics Education; Course: Mathematics and Physics Education; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: magistar/magistra edukacije matematike i fizike (magistar/magistra edukacije matematike i fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-06-17 11:33:32