Fractional Reaction-transport Equations Arising from Evanescent Continuous Time Random Walks

Continuous time random walks (CTRWs) describe a particular class of renewal processes used to model a wide variety of phenomena such as the motion of charge carriers in disordered systems, the dynamics of financial markets, the motion of diffusing particles in crowded environments, and certain anomalous relaxation phenomena in dielectric systems. It is well known that, on long time scales, a CTRW described by a separable probability density function (pdf) for the jump length of a particle and its waiting time between consecutive jumps yields a variety of fractional diffusion equations for suitable (and yet rather general) choices of both pdfs. Such fractional diffusion equations give rise to a broad range of behaviors for the mean square displacement of the particle, ranging from subdiffusive to superdiffusive. In this work we show that fractional equations with interesting solutions may also be obtained when the above CTRW is combined with a rather general class of death processes (evanescence), including a simple first-order process typical of radioactive decay. A general feature of the resulting fractional equations is that the parameters describing the decay process explicitly appear in the transport term, as opposed to heuristic fractional equations lacking a rigorous mesoscopic justification. In the subdiffusive case, we consider ∗E-mail address: [email protected] In: Fractional Calculus: Theory ISBN: 978-1-63463-002-3 Editors: R. Abi Zeid Daou and X. Moreau © 2015 Nova Science Publishers, Inc. 184 E. Abad, S. B. Yuste and and K. Lindenberg two applications of interest. First, we compute the survival probability of an immobile target surrounded by one or more subdiffusive traps that may spontaneously disappear in the course of their motion. The second example concerns a biologically relevant problem, namely, the formation of stationary morphogen concentration gradients by means of morphogen synthesis, anomalous diffusion, and degradation. PACS: 05.40.Jc, 82.20.Wt, 87.18.Pj