ABSTRACT
We analyze a set of d – dimensional fractional diffusion equations coupled with the reaction terms, which can be related to an irreversible or reversible process depending on the choice of the reaction terms. We consider the reaction terms for each equation given by R 1(ρ1, ρ2, t) = k 11ρ1(r, t) + k 12ρ2(r, t) and R 2(ρ1, ρ2, t) = k 21ρ1(r, t) + k 22ρ2(r, t), where the constants kij (i = 1, 2 and j = 1, 2) are chosen to define the reaction process. The differential operator connected to the time variable is extended to an arbitrary integrodifferential operator, which may be related to different operators with singular (Caputo operator) and nonsingular (Caputo-Fabrizio and Atangana-Baleanu operators) kernels. For the spatial operator, we consider an operator which may be related to different cases, particularly the Lévy distributions. This operator is a composition of the Lévy distribution with the generalized Hankel transform, which enables us to consider in a unified way different cases. We obtain the solutions for free boundary conditions and arbitrary initial conditions for this set of equations. They are obtained in terms of special functions such as the Mittag-Leffler and Fox H functions. We also analyze the behavior of the second moment and show that different regimes of diffusion can be obtained.
