Solution of fractional diffusion equations with absorbing boundaries

In this work we focus on the construction and solution of a fractional diffusion equation (FDE) in one-dimensional space with absorbing
boundaries. FDEs involve fractional derivatives in the form $d^\alpha/dx^\alpha$, with $\alpha \ \in \ \mathbb{R}$, enabling the
emergence of anomalous (superdiffusive) behavior which cannot be taken into account in the conventional (Brownian) diffusion equation. The
superdiffusive regime, in which we focus our work, arises from the solution of the FDE and also makes contact with the distribution of the
flight lengths of a L?vy flight random walker. By solving analytically the FDE we obtain the probability $W(x, t)$ of finding the walker in a
position $x$ at a given time $t$ in terms of Fox H-functions. We can also calculate the survival rate $S(t)$, which measures the probability that
the walker is still active (i.e. nonabsorbed) at a given time. We discuss the failure of the image method to solve the FDEs in a finite domain in
the context of the violation of the Sparre-Andersen theorem. We also propose an alternative approach to circumvent the problems related to the
images technique. In particular, the long-term behavior of the survival probability presents a time-dependence with a shift from the
(Spare-Andersen-like) power-law to the exponential decay. Our approach is based on both analytical as well as numerical techniques.