### Speaker

Prof.
Mamadsho Ilolov
(Center of Innovative Development of Science and New Technologies of Academy of Sciences of Tajikistan)

### Description

Classical description of diffusion process for scalar flux of particles of cosmic rays $u(\vec{r},t,E)$ is based on equation
$$
\frac{\partial u(\vec{r},t,E)}{\partial t}=k(E)u(\vec{r},t,E)+f(\vec{r},t,E),(1)
$$
where $E$ - energy of particles, $f(\vec{r},t,E)$ is density distribution of source and $k(E)$ - diffusion coefficient.
In the last years it is stated that energy spectrum of Galaxy cosmic rays falling to the Earth have a power type in wide diapason primary energy from $10^{9}eV$ until $10^{17}eV$ with "knee" at energy $\sim 3\cdot 10^{15}eV$ [1]. Phenomenon "knee" of spectrum it's beyond classical model of diffusion. In this relation we have to consider modified Fick's law which in its turn requires attraction of mathematical apparatus of fractional integro - differential calculations. Similar phenomenon is called an anomalous diffusion.
In this paper we consider fractional variant of equation (1)
\begin{equation}
D^{\beta}_{t}u(x,t,E)=\nabla^{{\frac{\alpha}{2}}}(k(x,t,E)\nabla^{{\frac{\alpha}{2}}}u(x,t,E))+f_{1}(x,t,E), x\in R^{3}, 0<\beta\leq 2,1<\alpha\leq 2.(2)
\end{equation}
Here $\alpha$ - fractional order of Caputo partial derivative in $x_{i}, i=1,2,3,$ $\beta$ - fractional order of Caputo derivative in $t$, $\nabla$ -gradient of function $u.$ In case $\alpha=2,\beta=1$ and k not depended on $x$ we get classical diffusion equation (1).
Numerical method of solution of initial - boundary problem for equation (2) is proposed, which elucidate the phenomenon of "knee" of spectrum.
REFERENCES
1. A.S.Borisov,V.M. Maximenko, V.S. Puchkov, S.E. Pyatovsky, S.A. Slavatinsky, A.V. Vargasov, R.A. Mukhamedshin. Some interesting phenomena observed in cosmic-ray experiments by means of X-ray emulsion technique at super accelerator energies , Physics of elementary particles and atomic nucleus 36,5(2005) 1227-1243.
2. P.Paradisi, R.Cesari, F.Mainardi, A.Maurizi, F.Tampieri. A generalized Fick's law to describe non-local transport effects. Physics and Chemistry of the Earth, Part B: Hydrology, Oceans and Atmosphere. 26,4 (2001) 275-279.

Registration number following "ICRC2015-I/" | 13 |
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Collaboration | -- not specified -- |

### Primary author

Prof.
Mamadsho Ilolov
(Center of Innovative Development of Science and New Technologies of Academy of Sciences of Tajikistan)

### Co-authors

Dr
Kholiknazar Kuchakshoev
(Russian-Tajik(Slavonic) University)
Mr
Shakarmamad Yormamadov
(International Scientific-Research Center "Pamir-Chakaltay")
Dr
Vitaly Puchkov
(Physical Institute of Russian Academy of Sciences)