Cattaneo-type subdiffusion-reaction equation

Subdiffusion in a system in which mobile particles $A$ can chemically react with static particles $B$ according to the rule $A+B\to B$ is considered within a persistent random-walk model. This model, which assumes a correlation between successive steps of particles, provides hyperbolic Cattaneo normal diffusion or fractional subdiffusion equations. Starting with the difference equation, which describes a persistent random walk in a system with chemical reactions, using the generating function method and the continuous-time random-walk formalism, we will derive the Cattaneo-type subdiffusion differential equation with fractional time derivatives in which the chemical reactions mentioned above are taken into account. We will also find its solution over a long time limit. Based on the obtained results, we will find the Cattaneo-type subdiffusion-reaction equation in the case in which mobile particles of species $A$ and $B$ can chemically react according to a more complicated rule.

Figure 1

Function Eq. (41) for various values of time $t$ given in the legend, $D_{\alpha }=0.01$, $\alpha =0.9$, $\beta =0.6$, $p=1.0$, ${\gamma }_1=0.5$, ${\gamma }_2=0.2$, $x_0=-1$, $x_r=0$ (all quantities are given in arbitrary chosen units).

Figure 2

Function Eq. (41) for various values of probability $\beta$ given in the legend, $t=2000$, the other parameters are the same as described in the legend of Fig. 1.