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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

${}_{x}{D}_{\theta }^{\alpha }u\left(x,t\right){=}_{t}{D}_{*}^{\beta }u\left(x,t\right),\phantom{\rule{1.em}{0ex}}x\in ℝ,\phantom{\rule{4pt}{0ex}}t\in {ℝ}^{+},\phantom{\rule{2.em}{0ex}}\left(1\right)$
$u\left(x,0\right)=\varphi \left(x\right),\phantom{\rule{1.em}{0ex}}x\in ℝ,\phantom{\rule{2.em}{0ex}}u\left(±\infty ,t\right)=0,\phantom{\rule{1.em}{0ex}}t>0,\phantom{\rule{2.em}{0ex}}\left(2\right)$

where $\varphi \in {L}^{c}\left(ℝ\right)$ is a sufficiently well-behaved function, ${}_{x}{D}_{\theta }^{\alpha }$ is the Riesz-Feller space-fractional derivative of the order $\alpha$ and the skewness $\theta$, and ${}_{t}{D}_{*}^{\beta }$ is the Caputo time-fractional derivative of the order $\beta$. If $1<\beta \le 2$, then the condition (2) is supplemented by the additional condition

${u}_{t}\left(x,0\right)=0·\phantom{\rule{2.em}{0ex}}\left(3\right)$

An analogon of the fundamental solution ${G}_{\alpha ,\beta }^{\theta }$ to the problem (1)–(2) (or (1)–(3)) is itroduced and determined via Fourier-Laplace transform:

$\stackrel{^}{\stackrel{˜}{{G}_{\alpha ,\beta }^{\theta }}}\left(\kappa ,s\right)=\frac{{s}^{\beta -1}}{{s}^{\beta }+{\psi }_{\alpha }^{\theta }\left(\kappa \right)},\phantom{\rule{2.em}{0ex}}\left(4\right)$

where

${\psi }_{\alpha }^{\theta }\left(\kappa \right)={|\kappa |}^{\alpha }{e}^{i\left(sign\phantom{\rule{0.166667em}{0ex}}\kappa \right)\theta \pi /2}·$

A scaling property as well as the similarity relation are obtained for ${G}_{\alpha ,\beta }^{\theta }$. 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 ($0<\alpha \le 2,\phantom{\rule{0.277778em}{0ex}}\beta =1$), time-fractional diffusion ($\alpha =2$, $0<\beta \le 2$) and neutral diffusion ($0<\alpha =\beta \le 2$). A composition rule for ${G}_{\alpha ,\beta }^{\theta }$ is established in the case $0<\beta \le 1$ 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 ${G}_{\alpha ,\beta }^{\theta }$ 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 pseudodifferential operators 26A33 Fractional derivatives and integrals (real functions) 33E12 Mittag-Leffler functions and generalizations 44A10 Laplace transform 35K05 Heat equation