Naslov (srp)

Numerička aproksimacija dvodimenzionih paraboličkih problema sa delta funkcijom

Autor

Sredojević, Bratislav, 1984-

Doprinosi

Bojović, Dejan, 1968-
Spalević, Miodrag, 1961-
Popović, Branislav, 1954-
Stanić, Marija, 1975-

Opis (eng)

Boundary problems for partial differential equations represent mathema- tical models of the most diverse phenomena, such as heat transfer, uid mechanics, atomic physics, etc. Only in rare cases, these tasks can be solved by classical methods of mathematical analysis, while in all other must be resort to approximate methods. Finite-difference method is one of the most commo- nly used methods for the numerical solution of boundary value problems for partial differential equations. In the context of nite-difference method, one of the main problems is proving convergence of difference schemes which appro- ximating boundary problems. Of particular interest are the estimates of the rate of convergence compatible with the smoothness of the coefficients and solution. When numerical approximations parabolic initial-boundary problems with generalized solutions, there are also some additional problems: the coefficients are not continuous functions, variable coefficients can be time-dependent coe- fficients and the solution belong to nonstandard anisotropic Sobolev spaces, etc. This dissertation is concerned with precisely these problems. The dissertation is considered a two-dimensional parabolic initial-boundary problem with concentrated capacity, that problem contains Dirac delta functi- on as the coefficient of the derivative by time. A further problem, in the case boundary problems with delta function as the coefficient, is that solution not in standard Sobolev spaces. The paper demonstrated a priori estimates of the corresponding non-standard norms. Assuming that the coefficients belong to anisotropic Sobolev spaces have been constructed the difference schemes with averaged right-hand side. The estimates of the rate of convergence in the spe- cial discrete fW2; 1 2 and fW1; 1=2 2 norms, is proved. The estimates of the rate of convergence compatible with the smoothness of the coefficients and solution, are obtained.

Opis (srp)

Granični problemi za parcijalne diferencijalne jednačine predsta- vljaju matematičke modele najraznovrsnijih pojava, kao na primer pro- voea toplote, mehanike fluida, procesa atomske fizike itd. Samo u retkim slučajevima ovi zadaci se mogu rexiti klasiqnim metodama ma- tematičke analize, dok se u svim ostalim mora pribegavati priblinim metodama. Metoda konaqnih razlika je jedan od najčešće primeiva- nih metoda za numeričko rešavanje graničnih problema za parcijalne diferencijalne jednačine. U okviru metode konačnih razlika, jedan od glavnih problema je dokazivanje konvergencije diferencijskih shema koje aproksimiraju granične probleme. Od posebnog interesa su ocene brzine konvergencije saglasne sa glatkošću koeficijenata i rešenja početnog problema. Prilikom numeričke aproksimacije poqetno-graničnih paraboliqkih problema sa generalisanim rešenjima javljaju se i neki dodatni pro- blemi: koeficijenti nisu neprekidne funkcije, promenljivi koefici- jenti mogu biti i vremenski zavisni, koeficijenti i rešenje pripadaju nestandardnim anizotropnim prostorima Soboljeva itd. Ova disertacija se upravo bavi tim problemima.

Jezik

srpski

Datum

2016

Licenca

Ovo delo je licencirano pod uslovima licence
Creative Commons CC BY-NC-ND 2.0 AT - Creative Commons Autorstvo - Nekomercijalno - Bez prerada 2.0 Austria License.

CC BY-NC-ND 2.0 AT

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