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Zbl 1054.35156
Mainardi, Francesco; Luchko, Yuri; Pagnini, Gianni
The fundamental solution of the space-time fractional diffusion equation. (English)
[J] Fract. Calc. Appl. Anal. 4, No. 2, 153-192 (2001). ISSN 1311-0454; ISSN 1314-2224/e

The authors study the Cauchy problem for the space-time fractional diffusion equation $$ _{x}D^{\alpha}_{\theta} u(x,t) = _{t}D^{\beta}_{\ast} u(x,t),\quad x\in {\Bbb R},\ t\in {\Bbb R}^{+}, \tag1$$ $$ u(x,0) = \varphi(x),\quad x\in {\Bbb R},\qquad u(\pm \infty, t) = 0,\quad t> 0,\tag2 $$ where $\varphi\in L^c({\Bbb R})$ is a sufficiently well-behaved function, $_{x}D^{\alpha}_{\theta}$ is the Riesz-Feller space-fractional derivative of the order $\alpha$ and the skewness $\theta$, and $_{t}D^{\beta}_{\ast}$ is the Caputo time-fractional derivative of the order $\beta$. If $1 < \beta \leq 2$, then the condition (2) is supplemented by the additional condition $$ u_t(x,0) = 0. \tag3$$ An analogon of the fundamental solution $G^{\theta}_{\alpha,\beta}$ to the problem (1)--(2) (or (1)--(3)) is itroduced and determined via Fourier-Laplace transform: $$ \widehat{\widetilde{G^{\theta}_{\alpha,\beta}}}(\kappa,s) = \frac{s^{\beta-1}}{s^{\beta} + \psi_{\alpha}^{\theta}(\kappa)}, \tag4$$ where $$ \psi_{\alpha}^{\theta}(\kappa) = |\kappa|^{\alpha} e^{i (sign\, \kappa) \theta\pi/2}. $$ A scaling property as well as the similarity relation are obtained for $G^{\theta}_{\alpha,\beta}$. It is found also the connection of the fundamental solution to the Mittag-Leffler function and to Mellin-Barnes integrals. Some particular cases are considered, namely space-fractional diffusion ($0 < \alpha \leq 2,\; \beta = 1$), time-fractional diffusion ($\alpha = 2$, $0 < \beta \leq 2$) and neutral diffusion ($0 < \alpha = \beta \leq 2$). A composition rule for $G^{\theta}_{\alpha,\beta}$ is established in the case $0 < \beta \leq 1$ which ensures its probabilistic interpretation at its range. A general representation of the Green function in terms of Mellin-Barnes integrals is obtained. On its base explicit formulas for $G^{\theta}_{\alpha,\beta}$ as well as asymptotics of the Green function for different values of the parameters are found. Qualita\-tive remarks concerning the solvability of the space-fractional diffusion equa\-tion are made illustrated by plots describing the behaviour of the Green function and the fundamental solution to (1).
[Sergei V. Rogosin (Minsk)]

MSC 2000:: *35S10 Initial value problems for pseudodifferential operators
26A33 Fractional derivatives and integrals (real functions)
33E12 Mittag-Leffler functions and generalizations
44A10 Laplace transform
35K05 Heat equation

Keywords: diffusion processes; fractional derivatives; integral transforms; Mittag-Leffler function; Mellin-Barnes integrals; Green function; stable probability distributions

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