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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:
\[ \widehat{\widetilde{G^{\theta}_{\alpha,\beta}}}(\kappa,s) = \frac{s^{\beta-1}}{s^{\beta} + \psi_{\alpha}^{\theta}(\kappa)}, \tag{4} \]
\[ \psi_{\alpha}^{\theta}(\kappa) = |\kappa|^{\alpha} e^{i (sign\, \kappa) \theta\pi/2}. \]
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).

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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