Similarity solutions for a class of Fractional Reaction-Diffusion equation

Abstract

This work studies exact solvability of a class of fractional reaction-diffusion equation with the Riemann-Liouville fractional derivatives on the half-line in terms of the similarity solutions. We derived the conditions for the equation to possess scaling symmetry even with the fractional derivatives. Relations among the scaling exponents are determined, and the appropriate similarity variable introduced. With the similarity variable we reduced the stochastic partial differential equation to a fractional ordinary differential equation. Exactly solvable systems are then identified by matching the resulted ordinary differential equation with the known exactly solvable fractional ones. Several examples involving the three-parameter Mittag-Leffler function (Kilbas-Saigo function) are presented. The models discussed here turn out to correspond to superdiffusive systems.

Figures (4)

• Figure 1: Plots of $y_j(z)$ with $\alpha=1/\beta, q=n+1-\beta$ for different $\beta$ (solid line for $\beta=2, n=j=1$) : (a) $\beta=1.9$ (dashed), $1.8$ (dot-dashed), and $1.7$ (dotted).; (b) $\beta=2.2$; (c) $\beta=2.4$, (d) $\beta=2.6$. For (b)-(d), $n=2$, $j=1$ (dashed), and $2$ (dotted). Number of terms used in the series expansion for the Kilbas-Saigo function is 400.
• Figure 2: Plots of $P(x,t)$ at different times for deformed Brownian-type diffusion, Eq. (\ref{['P1']}) with $q=2-\beta, \alpha = 1/\beta$, and $\beta= 2$ (solid), $1.8$ (dashed), and $1.7$ (dotted). Number of terms used in the series expansion for the Kilbas-Saigo function is 400.
• Figure 3: Plots of $P(x,t)$ at different times for deformed non-Brownian type diffusion Eq. (\ref{['P1']}) with $\alpha = 1, q=1\neq 2-\beta$, and $\beta= 2$ (solid), $1.9$ (dashed), and $1.7$ (dotted). Number of terms used in the series expansion for the Kilbas-Saigo function is 400.
• Figure 4: Plots of $P(x,t)$ at different times for deformed non-Fokker-Planck type reaction-diffusion Eq. (\ref{['P2']}) with $\mu_r=-0.5, \alpha = 1/\beta, q=2-\beta, c_1=c_2=1$, and $\beta= 2$ (solid), $1.9$ (dashed), and $1.8$ (dotted). Number of terms used in the series expansion for the Kilbas-Saigo function is 400.