Cauchy problem for fractional diffusion equations

Journal of Differential Equations

Volumn 199, Issue 2, 2004, Pages 211-255

  Eidelman, Samuil D ^a   Kochubei, Anatoly N ^a  

^a INSTITUTE OF MATHEMATICS   (Ukraine)

Author keywords

Fox's H function; Fractional derivative; Fractional diffusion equation; Fundamental solution; Levi method

Indexed keywords

EID: 1942542248     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2003.12.002     Document Type: Article

References (27)

1. V V. Anh, N.N. Leonenko, Spectral analysis of fractional kinetic equations with random data, J. Statist. Phys. 104 (2001) 1349-1387.
2. B. Baeumer, M. Meerschaert, Stochastic solutions for fractional Cauchy problems, Fract. Calc. Appl. Anal. 4 (2001) 481-500.
3. E. Bazhlekova, The abstract Cauchy problem for fractional evolution equation, Fract. Calc. Appl. Anal. 1 (1998) 255-270.
4. E. Bazhlekova, Fractional evolution equations in Banach spaces, Dissertation, Technische Universiteit Eindhoven, 2001.
5. B.L.J. Braaksma, Asymptotic expansions and analytic continuation for a class of Barnes integrals, Compositio Math. 15 (1964) 239-341.
6. M.M. Dzhrbashyan, A.B. Nersessyan, Fractional derivatives and Cauchy problem for differential equations of fractional order, Izv. AN Arm. SSR. Mat. 3 (1968) 3-29 (in Russian).
7. S.D. Eidelman, Parabolic Systems, North-Holland, Amsterdam, 1969.
8. A.M. El-Sayed, Fractional order evolution equations, J. Fract. Calc. 7 (1995) 89-100.
9. A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions, Vol. III, McGraw-Hill, New York, 1955.
10. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964.
11. R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Time fractional diffusion: a discrete random walk approach, Nonlinear Dynamics 29 (2002) 129-143.
12. A.N. Kochubei, A Cauchy problem for evolution equations of fractional order, Differential Equations 25 (1989) 967-974.
13. A.N. Kochubei, Fractional-order diffusion, Differential Equations 26 (1990) 485-492.
14. T. Kolsrud, On a class of probabilistic integrodifferential equations, in: S. Albeverio, H. Holden, J.E. Fenstad, T. Lindstrom (Eds.), Ideas and Methods in Mathematics and Physics. Memorial Volume Dedicated to Raphael Høegh-Krohn, Vol. 1, Cambridge University Press, Cambridge, 1992, pp. 168-172.
15. V.A. Kostin, Cauchy problem for an abstract differential equation with fractional derivatives, Russian Acad. Sci. Dokl. Math. 46 (1993) 316-319.
16. O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.
17. M.M. Meerschaert, D.A. Benson, H.P. Scheffler, B. Baeumer, Stochastic solutions of space-time fractional diffusion equations, Phys. Rev. E 65 (2002) 1103-1106.
18. R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000) 1-77.
19. K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
20. A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series, Vol. 3: More Special Functions, Gordon and Breach, New York, 1990.
21. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993.
22. W.R. Schneider, W. Wyss, Fractional Diffusion and wave equations, J. Math. Phys. 30 (1989) 134-144.
23. W.R. Schneider, Fractional diffusion, Lecture Notes Phys. 355 (1990) 276-286.
24. W.R. Schneider, Grey noise, in: Ideas and Methods in Mathematics and Physics. Memorial Volume Dedicated to Raphael Høegh-Krohn, Vol. 1, Cambridge University Press, Cambridge, 1992, pp. 261-282.
25. H.M. Srivastava, K.C. Gupta, S.P. Goyal, The H-Functions of One and Two variables with Applications, South Asian Publishers, New Dehli, 1982.
26. W. Wyss, The fractional diffusion equation, J. Math. Phys. 27 (1986) 2782-2785.
27. M. Yor, W. Schneider's grey noise and fractional Brownian motion, in: Proceedings of the Easter Meeting on Probability, Edinburgh, April 10-14, 1989.