Random death process for the regularization of subdiffusive anomalous equations

Abstract

Subdiffusive fractional equations are not structurally stable with respect to spatial perturbations to the anomalous exponent (Phys. Rev. E 85, 031132 (2012)). The question arises of applicability of these fractional equations to model real world phenomena. To rectify this problem we propose the inclusion of the random death process into the random walk scheme from which we arrive at the modified fractional master equation. We analyze the asymptotic behavior of this equation, both analytically and by Monte Carlo simulation, and show that this equation is structurally stable against spatial variations of anomalous exponent. Additionally, in the continuous and long time limit we arrived at an unusual advection-diffusion equation, where advection and diffusion coefficients depend on both the death rate and anomalous exponent. We apply the regularized fractional master equation to the problem of morphogen gradient formation.