Numerička aproksimacija dvodimenzionih paraboličkih problema sa delta funkcijom
Докторанд
Sredojević, BratislavМентор
Bojović, DejanЧланови комисије
Spalević, MiodragPopović, Branislav
Stanić, Marija
Метаподаци
Приказ свих података о дисертацијиСажетак
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, koef...icijenti i rešenje pripadaju
nestandardnim anizotropnim prostorima Soboljeva itd. Ova disertacija
se upravo bavi tim problemima.
Boundary problems for partial differential equations represent mathema-
tical models of the most diverse phenomena, such as heat transfer,
uid me-
chanics, 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 c...oe-
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.