$g$-fractional diffusion models in bounded domains

In the recent literature, the $g$-subdiffusion equation involving Caputo fractional derivatives with respect to another function has been studied in relation to anomalous diffusions with a continuous transition between different subdiffusive regimes. In this paper we study the problem of $g$-fractional diffusion in a bounded domain with absorbing boundaries. We find the explicit solution for the initial boundary value problem, and we study the first-passage time distribution and the mean first-passage time (MFPT). The main outcome is the proof that with a particular choice of the function $g$ it is possible to obtain a finite MFPT, differently from the anomalous diffusion described by a fractional heat equation involving the classical Caputo derivative.

Figure 1

First-passage time distributions for different choices of the function $g$. (a) The case of Caputo fractional derivative, $g\left(t\right)=t$. Dashed lines are asymptotic expressions (25), $f=A{t}^{-\delta }$ with $\delta =1+\alpha$. The MFPT is infinite for any value of $\alpha$. (b) The case of the Erdélyi-Kober derivative, $g\left(t\right)=t^{\beta }$ with $\beta =2$. The asymptotic behavior exponent is $\delta =1+\alpha \beta$. Only $\alpha >1/2$ corresponds to finite MFPT. (c) The case of the Hadamard fractional derivative, $g\left(t\right)=ln(t+1)$. The asymptotic behavior is $t^{-1}ln{\left(t\right)}^{-1-\alpha }$, between $t^{-1}$ and $t^{-2}$ (guide for eyes), implying divergent MFPT. (d) The case of exponential derivative $g\left(t\right)=exp\left(t\right)-1$. Asymptotic expressions are given by $Aexp(-\alpha t)$, and the MFPT is always finite. In all panels we report FPTD (full lines) calculated by numerically inverting the Laplace transform (23) and using (22) for three different values of the fractional derivative, $\alpha =0.9,0.6,0.3$. Dashed lines are asymptotic expressions (19). We set $x_0=0$, $a=b=1$, and $D=1$, so the prefactor in the asymptotic expressions is $A=-1/\left[2\mathrm{\Gamma }\right(-\alpha \left)\right]$.