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https://open.uns.ac.rs/handle/123456789/30177| Title: | Applications of Semigroups of Operators in Some Classes of Cauchy Problems Primene polugrupa operatora u nekim klasama Košijevih početnih problema |
Authors: | Žigić Milica | Keywords: | generalized stochastic process, chaos expansion,stochastic evolution equation, Wick product, white noise, C0−semigroup,infinitesimal generator, stochastic operator; fractional powers of operators,C-regularized resolvent families, abstract time-fractional equations.;uopšteni stohastički procesi, haos ekspanzija, stohastičke evolucione jednačine, Wick-ov proizvod, beli šum, C0−polugrupe, infinitezimalni generator, stohastički operator; frakcioni stepeni operatora, C-regularizovane rezolventne familije, apstraktne vremensko-frakcione jednačine. | Issue Date: | 22-Dec-2014 | Publisher: | Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu University of Novi Sad, Faculty of Sciences at Novi Sad |
Abstract: | <p>Doktorska disertacija je posvećena primeni teorije polugrupa operatora na rešavanje dve klase Cauchy-jevih početnih problema. U prvom delu smo<br />ispitivali parabolične stohastičke parcijalne diferencijalne jednačine (SPDJ-ne), odredjene sa dva tipa operatora: linearnim zatvorenim operatorom koji<br />generiše <em>C</em><sub>0</sub>−polugrupu i linearnim ograničenim operatorom kombinovanim<br />sa Wick-ovim proizvodom. Svi stohastički procesi su dati Wiener-Itô-ovom<br />haos ekspanzijom. Dokazali smo postojanje i jedinstvenost rešenja ove klase<br />SPDJ-na. Posebno, posmatrali smo i stacionarni slučaj kada je izvod po<br />vremenu jednak nuli. U drugom delu smo konstruisali kompleksne stepene<br /><em>C</em>-sektorijalnih operatora na sekvencijalno kompletnim lokalno konveksnim<br />prostorima. Kompleksne stepene operatora smo posmatrali kao integralne<br />generatore uniformno ograničenih analitičkih <em>C</em>-regularizovanih rezolventnih<br />familija, i upotrebili dobijene rezultate na izučavanje nepotpunih Cauchy-jevih problema viš3eg ili necelog reda.</p> <p>The doctoral dissertation is devoted to applications of the theory<br />of semigroups of operators on two classes of Cauchy problems. In the first<br />part, we studied parabolic stochastic partial differential equations (SPDEs),<br />driven by two types of operators: one linear closed operator generating a<br /><em>C</em><sub>0</sub>−semigroup and one linear bounded operator with Wick-type multipli-cation. All stochastic processes are considered in the setting of Wiener-Itô<br />chaos expansions. We proved existence and uniqueness of solutions for this<br />class of SPDEs. In particular, we also treated the stationary case when the<br />time-derivative is equal to zero. In the second part, we constructed com-plex powers of <em>C</em>−sectorial operators in the setting of sequentially complete<br />locally convex spaces. We considered these complex powers as the integral<br />generators of equicontinuous analytic <em>C</em>−regularized resolvent families, and<br />incorporated the obtained results in the study of incomplete higher or frac-tional order Cauchy problems.</p> |
URI: | https://open.uns.ac.rs/handle/123456789/30177 |
| Appears in Collections: | PMF Teze/Theses |
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