Nowa wersja platformy, zawierająca wyłącznie zasoby pełnotekstowe, jest już dostępna.
Przejdź na https://bibliotekanauki.pl

PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2008 | 6 | 3 | 737-753
Tytuł artykułu

Generalized Fokker-Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order α

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In a first stage, the paper deals with the derivation and the solution of the equation of the probability density function of a stochastic system driven simultaneously by a fractional Gaussian white noise and a fractional Poissonian white noise both of the same order. The key is the Taylor’s series of fractional order f(x + h) = E α(hαDx α)f(x) where E α() denotes the Mittag-Leffler function, and D x α is the so-called modified Riemann-Liouville fractional derivative which removes the effects of the non-zero initial value of the function under consideration. The corresponding fractional linear partial differential equation is solved by using a suitable extension of the Lagrange’s technique involving an auxiliary set of fractional differential equations. As an example, one considers a half-oscillator of fractional order driven by a fractional Poissonian noise.
Wydawca

Czasopismo
Rocznik
Tom
6
Numer
3
Strony
737-753
Opis fizyczny
Daty
wydano
2008-09-01
online
2008-07-17
Twórcy
autor
  • Department of Mathematics, University of Québec at Montréal, P.O. Box 8888, Downtown Station, Montréal, Qc H3C 3P8, Canada
Bibliografia
  • [1] N.U. Ahmed, C.D. Charalambous, SIAM J. Control Optim. 41, 313 (2002) http://dx.doi.org/10.1137/S0363012900368715[Crossref]
  • [2] V.V. Anh, N. Leonenko, Statisticsand Probability Letters 48, 239 (2000) http://dx.doi.org/10.1016/S0167-7152(00)00003-1[Crossref]
  • [3] V.V. Anh, N. Leonenko, Teor. Imovirnost. ta Matem. Statyst. 66, 3 (2002)
  • [4] E. Barkai, R. Metzler, J. Klafter, Phys. Rev. E 61, 132 (2000) http://dx.doi.org/10.1103/PhysRevE.61.132[Crossref]
  • [5] E. Barkai, Phys. Rev. E 63, 1 (2001) http://dx.doi.org/10.1103/PhysRevE.63.046118[Crossref]
  • [6] J. Beran, R. Sherman, M.S. Taqqu, W. Willinger, IEEE Trans. Commun. 43, 1566 (1995) http://dx.doi.org/10.1109/26.380206[Crossref]
  • [7] M. Caputo, Geophys. J.R. Ast. Soc. 13, 529 (1967)
  • [8] M.M. Djrbashian, S.S. Nersesian, Izv. AN Armjand. SSR 3, 3 (1968)
  • [9] C.A. Cornell, Stochastic process models in structural engineering, Technical report 34 (Departmentofcivil engineering, Stanford University, Stanford, CA, 1964)
  • [10] J.F. Crow, M. Kimura, An introduction to population geneticstheory (Harper and Row, 1970)
  • [11] W. Dai, C.C. Heyde, J. Appl. Math. Stochastic Anal. 9, 439 (1996) http://dx.doi.org/10.1155/S104895339600038X[Crossref]
  • [12] L. Decreusefond, A.S. Ustunel, Potential Anal. 10, 177 (1999) http://dx.doi.org/10.1023/A:1008634027843[Crossref]
  • [13] G. Del Frosso, A. Gerardi, F. Marchetti, Appl. Math. Optim. 7, 125 (1981) http://dx.doi.org/10.1007/BF01442110[Crossref]
  • [14] G. Doubleday, Mathematical Biosciences 17, 43 (1973) http://dx.doi.org/10.1016/0025-5564(73)90059-X[Crossref]
  • [15] T.E. Duncan, Y. Hu, B. Pasik-Duncan, SIAM J. Control Optim. 38, 582 (2000) http://dx.doi.org/10.1137/S036301299834171X[Crossref]
  • [16] A. El-Sayed, Int. J. Theor. Phys. 35, 311 (1996) http://dx.doi.org/10.1007/BF02083817[Crossref]
  • [17] A. El-Sayed, M.A.E. Aly, J. Appl. Math. & Computing 2, 525 (2002)
  • [18] I. Gikhman, A.W. Skorohod, Stochastic Differential Equations (Springer-Verlag, Berlin, 1972)
  • [19] M. Gregoriu, Applied Non-Gaussian Processes (Prentice-Hall, EnglewoodCliffs, NJ 1995)
  • [20] H. Haken, Information and self-organization. Amacro-scopic approach to complex systems, 3rd edition (Springer-Verlag, Berlin, 2006)
  • [21] A. Hanyga, P. Roy, Soc. Lond. A Mat. 458, 933 (2002) http://dx.doi.org/10.1098/rspa.2001.0904[Crossref]
  • [22] Y. Hu, B. Øksendal, Infinite Dim. Anal. Quantum Probab. Related Topics 6, 32 (2003).
  • [23] F. Huang, F. Liu, J. Appl. Math. & Computing 19, 190 (2005)
  • [24] K. Itô, Mem. Amer. Soc. 4 (1951)
  • [25] G. Jumarie, J. Math. Phys. 33, 3536 (1992) http://dx.doi.org/10.1063/1.529903[Crossref]
  • [26] G. Jumarie, Int. J. Syst. Sc. 6, 1113 (1993) http://dx.doi.org/10.1080/00207729308949547[Crossref]
  • [27] G. Jumarie, Systems Analysis, Modelling Simulation 45, 1483 (2002)
  • [28] G. Jumarie, Insurance: Mathematics and Economics 31, 179 (2002) http://dx.doi.org/10.1016/S0167-6687(02)00131-2[Crossref]
  • [29] G. Jumarie, Chaos. Soliton. Fract. 4, 907 (2004) http://dx.doi.org/10.1016/j.chaos.2004.03.020[Crossref]
  • [30] G. Jumarie, Appl. Math. Lett. 18, 739 (2005) http://dx.doi.org/10.1016/j.aml.2004.05.014[Crossref]
  • [31] G. Jumarie, Appl. Math. Lett. 18, 817 (2005) http://dx.doi.org/10.1016/j.aml.2004.09.012[Crossref]
  • [32] G. Jumarie, Journal of Applied Mathematics and Informatics (inpress)
  • [33] H. Kober, Quart. J. Math. Oxford 11, 193 (1940) http://dx.doi.org/10.1093/qmath/os-11.1.193[Crossref]
  • [34] A.N. Kochubei, J. Diff. Eqs. 26, 485 (1990)
  • [35] K.M. Kolwankar, A.D. Gangal, Pramana-J. Phys. 48, 49 (1997) http://dx.doi.org/10.1007/BF02845622[Crossref]
  • [36] K.M. Kolwankar, A.D. Gangal, Phys. Rev. Lett. 80, 214 (1998) http://dx.doi.org/10.1103/PhysRevLett.80.214[Crossref]
  • [37] A.V. Letnivov, Math. Sb. 3, 1 (1968)
  • [38] Y.K. Lin, J. Appl. Mech. ASME 30, 555 (1973)
  • [39] S.J. Lin, Stochastics Rep. 55, 121 (1995)
  • [40] J. Liouville, J. Ecole Polytechnique 13, 71 (1832)
  • [41] F. Mainardi, Y. Luchko, G. Pagnini, Fractional Calculus and Applied Analysis 4, 153 (2001)
  • [42] B.B. Mandelbrot, J.W. van Ness, SIAM Rev. 10, 422 (1968) http://dx.doi.org/10.1137/1010093[Crossref]
  • [43] B.B. Mandelbrot, R. Cioczek-Georges, Stochastic Processes and their Applications 60, 1 (1995) http://dx.doi.org/10.1016/0304-4149(95)00046-1[Crossref]
  • [44] B.B. Mandelbrot, R. Cioczek-Georges, Stochastic Processes and their Applications 64, 143 (1996) http://dx.doi.org/10.1016/S0304-4149(96)00089-0[Crossref]
  • [45] B.B. Mandelbrot, Fractals and scaling in finance: Discontinuity, concentration, risk (Springer-Verlag, NewYork,Berlin, 1997)
  • [46] S.I. Muslih, D. Baleanu, E.M. Rabei, Cent. Eur. J. Phys. 5, 549 (2007) http://dx.doi.org/10.2478/s11534-007-0041-6[Crossref]
  • [47] T.J. Osler, SIAM. J. Mathematical Analysis 2, 37 (1971) http://dx.doi.org/10.1137/0502004[Crossref]
  • [48] I. Podlubny, Fractional differential equations (Academic Press, San Diego, 1999)
  • [49] A. Ramakrishman, S.K. Srinivasan, Bull. Math. Biophys 20, 289 (1930) http://dx.doi.org/10.1007/BF02477887[Crossref]
  • [50] J.B. Roberts, J. Sound Vibration 24, 23 (1972) http://dx.doi.org/10.1016/0022-460X(72)90119-8[Crossref]
  • [51] A. Saichev, G. Zaslavsky, Chaos 7, 753 (1997) http://dx.doi.org/10.1063/1.166272[Crossref]
  • [52] G. Samorodnitski, M. Grigoriu, Teoret. Mat. Fiz. 3, 391 (2007), translation in Theor. Math. Phys. 3, 332 (2007)
  • [53] W.R. Schneider, W. Wyss, J. Math. Phys. 30, 134 (1989) http://dx.doi.org/10.1063/1.528578[Crossref]
  • [54] N.T. Shawagfeh, Appl. Math. And Comp. 131, 517 (2002) http://dx.doi.org/10.1016/S0096-3003(01)00167-9[Crossref]
  • [55] R.L. Stratonovich, SIAM J. Control Optim. 4, 362 (1966) http://dx.doi.org/10.1137/0304028[Crossref]
  • [56] B. Tsybakov, N. Georganas, IEEE Trans. Inform.Theory 44, 1713 (1998) http://dx.doi.org/10.1109/18.705538[Crossref]
  • [57] C.C. Tung, J. Engrg. Mech. ASCE 93, 79 (1967)
  • [58] S.F. Wojtkiewicz, E.A. Johnson, L.A. Bergman, M. Grigoriu, B.F.Jr Spencer, Comput. Method. Appl. M. 168, 73 (1999) http://dx.doi.org/10.1016/S0045-7825(98)00098-X[Crossref]
  • [59] W. Wyss, Fract. Calc. Appl. Anal. 3, 51 (2000)
  • [60] M. Zahle, Probability theory and related fields 111, 333 (1998) http://dx.doi.org/10.1007/s004400050171[Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.-psjd-doi-10_2478_s11534-008-0090-5
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.