Vol.4, No 16, 2004 pp. 11-31
UDC 519.218.7:531.36:629.7.058.6
Invited Paper
SOME ANALYTIC ITERATIVE METHODS FOR SOLVING
VARIOUS CLASSES OF STOCHASTIC HEREDITARY INTEGRODIFFERENTIAL EQUATIONS
Svetlana Janković, Miljana Jovanović
Faculty of Science, Department of Mathematics, University of Niš,
Višegradska 33, 18000 Niš, Serbia and Montenegro
E-mail: svjank@pmf.ac.ni.yu,
mima@pmf.ac.ni.yu
Abstract. The notion of hereditary phenomena is particularly convenient
for studying such phenomena in continuum mechanics of materials with memories,
as a version of the well-known theory of fading memory spaces. Mathematical
models represent deterministic hereditary differential equations researched
in many papers and monographs. Later, this notion was appropriately used
in an investigation of the effect of the Gaussian white noise, which mathematical
interpretation is represented by stochastic hereditary differential equations
of the Ito type.
In the present paper we consider a general analytic iterative method
for solving stochastic hereditary integrodifferential equation of the Ito
type. We give sufficient conditions under which a sequence of iterations
converges with probability one to the solution of the original equation.
The generality of this method is in the sense that many well-known iterative
methods are its special cases, the Picard-Lindelof method of successive
approximations, for example. Some other iterative methods, including linearizations
of the coefficients of the original equation, are suggested.
Especially, using a concept of a random bounded integral contractor,
basically introduced by Altman and Kuo, we show that the iterative procedure
utilized to prove the existence and uniqueness of the solution of the stochastic
hereditary integrodifferential equation, is also a special algorithm included
in the considered general iterative procedure.
Key words: Stochastic differential equation, stochastic
hereditary integrodifferential equation, random integral contractor,
Z-algorithm, determining sequence.
NEKE ANALITIČKE ITERATIVNE METODE ZA REŠAVANJE
RAZLIČITIH KLASA STOHASTIČKIH NASLEDNIH INTEGRODIFERENCIJALNIH JEDNAČINA
Nasledni fenomeni su posebno pogodni za proučavanje fenomena u mehanici
kontinuuma materijala sa memorijom. Matematički modeli takvih pojava se
opisuju determinističkim naslednim diferencijalnim jednačinama, proučavanim
u mnogim radovima i monografijama. Kasnije, ovi pojmovi su adekvatno prošireni
na istraživanja pod uticajem Gaussovog belog šuma, sa matematičkom interpretacijom
stohastičkim naslednim diferencijalnim jednačinama tipa Itoa.
U ovom radu se razmatra opšta analitička iterativna metoda za rešavanje
stohastičkih naslednih integrodiferencijalnih jednačina tipa Itoa. Daju
se dovoljni uslovi pri kojima niz iteracija konvergira u verovatnoći ka
rešenju originalne jednačine. Ova metoda je opšta, u smislu da su mnoge
poznate iterativne metode njeni specijalni slučajevi, na primer metoda
sukcesivnih aproksimacija Picard-Lindelofa. Prikazane su i neke druge iterativne
metode sa linearizacijom koeficijenata originalne jednačine.
Specijalno, koristeći koncept ograničenog slučajnog integralnog kontraktora
u smislu Altmana i Kuoa, pokazuje se da je iterativna metoda primenjena
u dokazu teoreme egzistencije i jedinstvenosti rešenja stohastičke nasledne
integrodiferencijalne jednačine takodje specijalan slučaj prethodno opisane
opšte iterativne metode.
