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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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