Disipativna Burgersova jednadžba

master's thesis

• Author: Ivona Karmelić
• Mentor: Siniša Slijepčević (mentor)
• Committee member: Siniša Slijepčević (predsjednik povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Josip Tambača (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-07-17, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu smo promatrali nelinearnu disipativnu Burgersovu jednadžbu \[ u_{t}+uu_{x}=\varepsilon u_{xx}. \] U Poglavlju 1 promatrali smo prostore funkcija. Definirali smo Banachove i Hilbertove prostore i naveli važne primjere tih prostora. U Poglavlju 2 definirali smo posebnu klasu linearnih operatora na Banachovim prostorima, takozvane sektorijalne operatore, i pokazali neka njihova korisna svojstva. U Poglavlju 3 promatrali smo nelinearni problem \begin{align} \frac{du}{dt}+Au&=f(t,\,u),\ t>t_{0},\nonumber \\ u(t_{0})&=u_{0},\nonumber \end{align} pri čemu je \(A\) bio sektorijalan operator i \(f\) lokalno Hölder neprekidna po \(t\) i lokalno Lipschitz neprekidna po \(u\). Naveli smo i dokazali nužne uvjete za egzistenciju i jedinstvenost rješenja gornjeg problema. U Poglavlju 4 smo promatrali inicijalni problem za disipativnu Burgersovu jednadžbu. Pokazali smo da za operator \(A=\varepsilon u_{xx}\) i funkciju \(f=uu_{x}\) vrijede pretpostavke iz Poglavlja 3 i time dokazali da inicijalni problem za disipativnu Burgersovu jednadžbu ima jedinstveno rješenje. Jednadžbu smo linearizirali Hopf - Cole transformacijom kako bi dobili eksplicitnu formulu za rješenje.

Abstract (english)

In this diploma thesis we analysed nonlinear dissipative Burgers’ equation \[ u_{t}+uu_{x}=\varepsilon u_{xx}. \] In Chapter 1 we discussed function spaces. We defined Banach and Hilbert spaces and gave examples of those spaces. In Chapter 2 we defined a special class of linear operators on Banach spaces, so-called sectorial operators, and showed some of their properties. In Chapter 3 we considered a nonlinear problem \begin{align} \frac{du}{dt}+Au&=f(t,\,u),\ t>t_{0},\nonumber \\ u(t_{0})&=u_{0},\nonumber \end{align} where \(A\) was a sectorial operator and \(f\) locally Hölder continuous in \(t\) and locally Lipschitz in \(u\). We showed necessary conditions for existence and uniqueness of the solution. In Chapter 4 we considered the initial problem for dissipative Burgers’ equation. We showed that operator \(A=\varepsilon u_{xx}\)and function \(f=uu_{x}\)satisfy presumptions from chapter 3 and proved that initial problem for dissipative Burgers’ equation has a unique solution. We linearized the equation using the Hopf - Cole transform to get an explicit formula for the solution.

• Keywords: nelinearna disipativna Burgersova jednadžba; Banachovi prostori; Hilbertovi prostori; sektorijalni operatori; Hopf - Cole transformacija
• Keywords (english): nonlinear dissipative Burgers’ equation; Banach spaces; Hilbert spaces; linear operators on Banach spaces; sectorial operators; Hopf - Cole transform
• Language: croatian
• URN:NBN: urn:nbn:hr:217:475748
• Study programme: Title: Applied Mathematics; 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: 2018-01-31 11:51:16