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The fundamental solutions of the time-fractional diffusion equation. (English) Zbl 1068.35038

Fabrizio, Mauro (ed.) et al., Mathematical models and methods for smart materials. Proceedings of the conference, Cortona, Italy, June 25–29, 2001. River Edge, NJ: World Scientific (ISBN 981-238-235-6/hbk). Ser. Adv. Math. Appl. Sci. 62, 207-224 (2002).
The time-fractional diffusion equation considered here is \[ \frac{\partial^\beta}{\partial t^\beta}\, u(x,t)=\frac{\partial^2}{\partial x^2}\, u(x,t),\quad 0<\beta\leq 2,\;x\in\mathbb{R},\;t\in\mathbb{R}^+_0, \] where \(\beta\) is not an integer \((\beta\neq 1,2)\) and \[ \frac{\partial^\beta}{\partial t^\beta}\, u(x,t)= \begin{cases}\frac1{\Gamma(1-\beta)}\int^t_0 \left[\frac\partial{\partial\zeta}\,u(x,\zeta)\right]\;\frac{d\zeta}{(t-\zeta)}\,\beta,\quad 0<\beta<1,\\ \frac1{\Gamma(2-\beta)}\int^t_0\left[\frac{\partial^2}{\partial\zeta^2}\,u(x,\zeta)\right]\;\frac{d\zeta}{(t-\zeta)^{\beta-1}}\,\beta,\quad 1<\beta<2.\end{cases} \] The Fourier and Laplace transforms with respect to the space and time variables are used here to obtain the fundamental solutions of these equations. The general representation of fundamental solutions (the reduced Green’s functions) are expressed in terms of Fox’s \(H\)-functions.
Since the \(H\)-functions are computable, the graphs of the reduced Green’s functions are presented in a number of cases.
For the entire collection see [Zbl 1021.00008].

MSC:

35K15 Initial value problems for second-order parabolic equations
26A33 Fractional derivatives and integrals
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35A08 Fundamental solutions to PDEs
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