Fractional equation approaches to subdiffusion problems: a tale of tails
Enrique Abad, Dpto. de Física Aplicada (Univ. de Extremadura)
Anomalous transport (in the sense of a nonlinear growth of the mean square displacement) is ubiquitous in nature, notably in biological systems. A widely used model to mimic anomalous transport processes with a large dispersion of waiting times is the celebrated Continuous Time Random Walk (CTRW). The CTRW with a long-tailed waiting time distribution and a jump length distribution of finite variance (both decoupled from one another) is known to become equivalent to a fractional diffusion equation in the long-time limit. While the fractional diffusion equation is an integrodifferential equation, it is nevertheless amenable to exact solution via Laplace transform methods or a variable separation ansatz.
We consider a number of first-passage problems of interest involving the solution of the fractional diffusion equation with absorbing boundary conditions. The solutions are characterized by a very detailed memory of the initial condition persisting even in the long-time regime; the solutions are also peculiar from a mathematical viewpoint as a) the long-time decay modes can be used to construct new polynomial approximations for Bessel functions and b) for a proper choice of the system parameters divergent series are seen to emerge in the solution. Such series must be suitably regularized to recover the physically correct solutions.
Finally, we shall also give an overview on how to deal with problems where the absorption process is not constrained to a boundary, but is delocalized in space. In this case one must use highly non-intuitive reaction-subdiffusion equations which find a proper justification in the framework of a mesoscopic approach. We shall discuss the application of this type of equations to understand a key process in developmental biology, namely, the formation of morphogen concentration gradients by means of subdiffusive transport.