## Cauchy problem for fractional diffusion equations.(English)Zbl 1068.35037

Equations of the form $(D^{(\alpha)}_tu)(t,x)-B u(t,x)=f(t,x),\quad t\in[0,\tau], \quad 0<\alpha<1,\;x\in\mathbb R^n$ where $(D^{(\alpha)}_tu)(t,x)= \frac1{\Gamma(1-2)}\left[\frac\partial{\partial t}\int^t_0(t-\zeta)^{-\alpha}u(\zeta,x)\,d\zeta-t^{-\alpha}u(0,x)\right]$
$B= \sum^n_{k,j=1} a_{ij}(x)\frac{\partial^2}{\partial x_i\partial x_j}+\sum^n_{j=1}b_j(x)\frac{\partial}{\partial x_j}+c(x)$ are considered here. The fundamental solution is studied via a Green matrix. The arguments of the Green matrix are expresssed in terms of Fox’s $$H$$-functions. Estimates of the elements of the Green matrix are also presented.

### MSC:

 35K15 Initial value problems for second-order parabolic equations 26A33 Fractional derivatives and integrals

### Keywords:

Fox’s $$H$$-function; fundamental soloution
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### References:

 [1] V. Anh, V; Leonenko, N.N., Spectral analysis of fractional kinetic equations with random data, J. statist. phys., 104, 1349-1387, (2001) · Zbl 1034.82044 [2] Baeumer, B.; Meerschaert, M., Stochastic solutions for fractional Cauchy problems, Fract. calc. appl. anal., 4, 481-500, (2001) · Zbl 1057.35102 [3] Bazhlekova, E., The abstract Cauchy problem for fractional evolution equation, Fract. calc. appl. anal., 1, 255-270, (1998) · Zbl 1041.34043 [4] E. Bazhlekova, Fractional evolution equations in Banach spaces, Dissertation, Technische Universiteit Eindhoven, 2001. [5] Braaksma, B.L.J., Asymptotic expansions and analytic continuation for a class of Barnes integrals, Compositio math., 15, 239-341, (1964) · Zbl 0129.28604 [6] Dzhrbashyan, M.M.; Nersessyan, A.B., Fractional derivatives and Cauchy problem for differential equations of fractional order, Izv. AN arm. SSR. mat., 3, 3-29, (1968), (in Russian) [7] Eidelman, S.D., Parabolic systems, (1969), North-Holland Amsterdam [8] El-Sayed, A.M., Fractional order evolution equations, J. fract. calc., 7, 89-100, (1995) · Zbl 0839.34069 [9] A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions, Vol. III, McGraw-Hill, New York, 1955. · Zbl 0064.06302 [10] Friedman, A., Partial differential equations of parabolic type, (1964), Prentice-Hall Englewood Cliffs, NJ · Zbl 0144.34903 [11] Gorenflo, R.; Mainardi, F.; Moretti, D.; Paradisi, P., Time fractional diffusiona discrete random walk approach, Nonlinear dynamics, 29, 129-143, (2002) · Zbl 1009.82016 [12] Kochubei, A.N., A Cauchy problem for evolution equations of fractional order, Differential equations, 25, 967-974, (1989) · Zbl 0696.34047 [13] Kochubei, A.N., Fractional-order diffusion, Differential equations, 26, 485-492, (1990) · Zbl 0729.35064 [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] Kostin, V.A., Cauchy problem for an abstract differential equation with fractional derivatives, Russian acad. sci. dokl. math., 46, 316-319, (1993) [16] Ladyzhenskaya, O.A.; Solonnikov, V.A.; Uraltseva, N.N., Linear and quasilinear equations of parabolic type, (1968), American Mathematical Society Providence, RI · Zbl 0174.15403 [17] Meerschaert, M.M.; Benson, D.A.; Scheffler, H.P.; Baeumer, B., Stochastic solutions of space-time fractional diffusion equations, Phys. rev. E, 65, 1103-1106, (2002) [18] Metzler, R.; Klafter, J., The random Walk’s guide to anomalous diffusiona fractional dynamics approach, Phys. rep., 339, 1-77, (2000) · Zbl 0984.82032 [19] Miller, K.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002 [20] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series, Vol. 3: More Special Functions, Gordon and Breach, New York, 1990. · Zbl 0967.00503 [21] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach New York · Zbl 0818.26003 [22] Schneider, W.R.; Wyss, W., Fractional diffusion and wave equations, J. math. phys., 30, 134-144, (1989) · Zbl 0692.45004 [23] Schneider, W.R., Fractional diffusion, Lecture notes phys., 355, 276-286, (1990) · Zbl 0721.60086 [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] Srivastava, H.M.; Gupta, K.C.; Goyal, S.P., The H-functions of one and two variables with applications, (1982), South Asian Publishers New Dehli · Zbl 0506.33007 [26] Wyss, W., The fractional diffusion equation, J. math. phys., 27, 2782-2785, (1986) · Zbl 0632.35031 [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.
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