O nekim transmisionim problemima u disjunktnim oblastima

Abstract

U primenama, naročito u inženjerstvu, često se sreću kompozitne ili slojevite strukture, pri čemu se osobine pojedinih slojeva mogu značajno razlikovati od osobina okolnog materijala. Slojevi mogu imati strukturnu, termičku, elektromagnetsku ili optičku ulogu itd. Matematičkim modelovanjem prenosa energije i mase u oblastima sa slojevima dobijaju se tzv. transmisioni problemi. Na samom početku, u disertaciji se razmatra transmisioni spektralni problem u oblasti koja se sastoji od dva disjunktna intervala. Na svakom intervalu zadat je problem sopstvenih vrednosti, dok se interakcija između njihovih rešenja opisuje nelokalnim uslovima saglasnosti. Dokazana je egzistencija prebrojivog niza generalisanih rešenja, pri čemu uređeni parovi sopstvenih funkcija pripadaju odgovarajućim prostorima Soboljeva. Opisana je struktura spektra i asimptotsko ponašanje sopstvenih vrednosti. Konstruisana je diferencijska shema za njihovo rešavanje. Pored transmisionog spektralnog problema, u disertaciji se razmatraju klase nestandardnih eliptičkih i paraboličkih transmisionih problema u disjunktnim oblastima. Kao modelni primer uzeta je oblast koja se sastoji iz dva nesusedna pravougaonika. U svakoj podoblasti zadat je granični problem eliptičkog tipa, odnosno početno-granični problem paraboličkog tipa. Interakcija između rešenja opisuje se pomoću nelokalnih uslova saglasnosti na granicama posmatranih podoblasti. Razmotreno je više primera žičkih i inženjerskih zadataka koji se svode na transmisione probleme sličnog tipa. Za modelne probleme dokazana je egzistencija i jedinstvenost rešenja u odgovarajućim prostorima Soboljeva. Takođe su konstruisane diferencijske sheme za njihovo rešavanje i dokazana njihova konvergencija.

In applications, especially in engineering, often are encountered composite or layered structures, where the properties of individual layers can vary considerably from the properties of the surrounding material. Layers can be structural, thermal, electromagnetic or optical, etc. Mathematical models of energy and mass transfer in domains with layers lead to so called transmission problems. At the beginning, in this dissertation we consider a transmission spectral problem in the area, which consists of two disjoint intervals. At each interval was given a eigenvalue problem, while the interaction between their solutions is described by means of the nonlocal integral conjugation conditions. The existence of countable series of generalized solutions is proved, whereby ordered pairs of eigenfunctions belong to the corresponding Sobolev spaces. The structure of the spectrum and asymptotic behavior of eigenvalues is described. A diference scheme for solving them is constructed. In addition to the spectral transmission problems, in this dissertation we consider a class of non-standard elliptic and parabolic transmission problems in disjoint domains. As a model example it is taken an area consisting of two non-adjacent rectangles. In each subarea was given a boundary problem of elliptic type, as well as initial-boundary problem of the parabolic type. The interaction between their solutions is described by means of the nonlocal integral conjugation conditions. It was considered more examples of physical and engineering tasks which are reduced to transmission problems of similar type. For the model problems the existence and uniqueness of its weak solution in appropriate Sobolev-like space is proved. They are also constructed a finite diference schemes for solving them and proved their convergence.