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The fundamental solution of the space-time fractional diffusion equation. (English) Zbl 1054.35156
The authors study the Cauchy problem for the space-time fractional diffusion equation
$_{x}D^{\alpha}_{\theta} u(x,t) = _{t}D^{\beta}_{\ast} u(x,t),\quad x\in {\mathbb R},\;t\in {\mathbb R}^{+}, \tag{1}$
$u(x,0) = \varphi(x),\quad x\in {\mathbb R},\qquad u(\pm \infty, t) = 0,\quad t> 0,\tag{2}$
where $$\varphi\in L^c({\mathbb R})$$ 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
$u_t(x,0) = 0. \tag{3}$
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:
$\widehat{\widetilde{G^{\theta}_{\alpha,\beta}}}(\kappa,s) = \frac{s^{\beta-1}}{s^{\beta} + \psi_{\alpha}^{\theta}(\kappa)}, \tag{4}$
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. Qualitative remarks concerning the solvability of the space-fractional diffusion equation are made illustrated by plots describing the behaviour of the Green function and the fundamental solution to (1).

##### MSC:
 35S10 Initial value problems for PDEs with pseudodifferential operators 26A33 Fractional derivatives and integrals 33E12 Mittag-Leffler functions and generalizations 44A10 Laplace transform 35K05 Heat equation
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