Harmonic analysis of random fractional diffusion-wave equations

Applied Mathematics and Computation

Volumn 141, Issue 1, 2003, Pages 77-85

  Anh, V V ^a   Leonenko, N N ^b,c  

^a QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^b CARDIFF UNIVERSITY   (United Kingdom)

^c TARAS SHEVCHENKO NATIONAL UNIVERSITY OF KYIV   (Ukraine)

Author keywords

Diffusion wave equation with random data; Fractional diffusion equation; Mittag Leffler function; Spectral representation; Volterra type fractional integral equation

Indexed keywords

DIFFUSION; GREEN'S FUNCTION; HARMONIC ANALYSIS; INTEGRAL EQUATIONS;

MITTAG-LEFFLER FUNCTION;

WAVE EQUATIONS;

EID: 0037457782     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0096-3003(02)00322-3     Document Type: Conference Paper

References (26)

1. Angulo J.M., Ruiz-Medina M.D., Anh V.V., Grecksch W. Fractional diffusion and fractional heat equation. Adv. Appl. Prob. 32(4):2000;1077-1099.
2. Anh V.V., Leonenko N.N. Non-Gaussian scenarios for the heat equation with singular initial data. Stoch. Proc. Appl. 84:1999;91-114.
3. Anh V.V., Leonenko N.N. Scaling laws for fractional diffusion-wave equation with singular initial data. Stat. Prob. Lett. 48:2000;239-252.
4. V.V. Anh, N.N. Leonenko, Spectral analysis of fractional kinetic equations with random data, J. Stat. Phys., vol. 104(5/6) (2001) 1349-1387.
5. Beghin L., Knopova V.P., Leonenko N.N., Orsingher E. Gaussian limiting behaviour of the rescaled solution to the linear Korteweg-de Vries equation with random initial conditions. J. Stat. Phys. 99(3/4):2000;769-781.
6. Caputo M. Elasticità e Dissipazione. 1969;Zanichelli, Bologna.
7. de Feriet J.K. Random solutions of the partial differential equations. Proc. 3rd Berkeley Symp. Math. Stat. Prob., vol. III. 1955;199-208 University of California Press, Berkeley, CA.
8. Djrbashian M.M. Harmonic Analysis and Boundary Value Problems in Complex Domain. 1993;Birkhäuser Verlag, Basel.
9. Djrbashian M.M., Nersesian A.B. Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. Acad. Nauk Armjanskvy SSR. 3(1):1968;3-29. (In Russian).
10. Gorenflo R., Iskenderov A., Luchko Y. Mapping between solutions of fractional diffusion-wave equations. Fract. Calculus Appl. Anal. 3(1):2000;75-86.
11. Gorenflo R., Luchko Y., Mainardi F. Analytical properties and applications of the Wright function. Fract. Calculus Appl. Anal. 2:1999;383-414.
12. Gorenflo R., Luchko Y., Mainardi F. Wright functions and scale-invariate solutions of the diffusion-wave equation. J. Comput. Appl. Math. 118:2000;175-191.
13. Kochubei A.N. Fractional order diffusion. J. Diff. Eqs. 26(4):1990;485-492.
14. Leonenko N. Limit Theorems for Random Fields with Singular Spectrum. 1999;Kluwer.
15. Luchko Y., Gorenflo R. An operational method for solving fractional differential equations with the Caputo derivatives. Acta Mathematica Vietnamica. 24:1999;207-233.
16. Mainardi F. The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9(6):1996;23-28.
17. Mainardi F., Gorenflo R. On Mittag-Leffler type functions in fractional evolution processes. J. Comput Appl. Math. 118:2000;283-299.
18. F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calculus Appl. Anal. 4 (2) (2001) 153-192.
19. Millet A., Morien P.-L. On a stochastic wave equation in two space dimensions: regularity of the solution and its density. Stoch. Proc. Appl. 86:2000;141-162.
20. Orsingher E. Damped vibrations excited by white noise. Adv. Appl. Prob. 16(3):1984;562-584.
21. Peszat S., Zabczyk J. Nonlinear stochastic wave and heat equations. Prob. Theory Relat. Fields. 116:2000;421-443.
22. Podlubny I. Fractional Differential Equations. 1999;Academic Press, San Diego, CA.
23. Saichev A.I., Zaslawsky G.M. Fractional kinetic equations: solutions and applications. Chaos. 7(4):1997;753-764.
24. W.R. Schneider, Fractional diffusion, in: R. Lima, L. Streit, D.V. Mendes (Eds.), Dynamics and Stochastic Processes, Theory and Applications, Volume 355 of Lecture Notes in Physics, Springer, Heidelberg, 1990, pp. 276-286.
25. Schneider W.R., Wyss W. Fractional diffusion and wave equations. J. Math. Phys. 30(1):1989;134-144.
26. Srivastava H.M., Gupta K.C., Goyal S.P. The H-functions of one and two variables with applications. 1982;South Asia Publishers, New Delhi, Madras.