On Hilfer's objection to the fractional time diffusion equation |
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| Institution: | 1. Department of Physics, Bar Ilan University, Ramat-Gan 52900 Israel;2. Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin Germany;1. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing, China;2. Department of Bioengineering, University of Illinois at Chicago, Chicago, IL, United States;1. ACRIM, Coronado, CA 92118, USA;2. Duke University, Durham, NC 27708, USA;1. Grupo de Estudios Ambientales, Universidad Tecnológica Nacional, Colón 332, San Nicolás (2900), Bs.As., Argentina;2. Facultad de Ingeniería, Universidad Nacional de Entre Ríos, Ruta Prov. 11 km Oro Verde (3100), Paraná, Entre Ríos, Argentina;1. Grupo de Estudios Ambientales, Universidad Tecnológica Nacional, Colón 332, San Nicolás (2900), Bs.As., Argentina;2. Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA |
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| Abstract: | Hilfer Physica A 329 (2003) 35] claims to give an example of a continuous time random walk (CTRW) model with long-tailed waiting time probability density that approaches a Gaussian behavior in the continuum limit. Rigorous limit theorems, derived previously, show however that in the limit of long-time such a CTRW converges to a non-Gaussian behavior. We discuss two types of continuum limits for the CTRW model: the fractional continuum limit and the one introduced by Hilfer. We show that the fractional limit yields the correct long-time behavior of the CTRW, while Hilfer's continuum limit does not. We discuss a general approach to find a continuum limit of the CTRW process. |
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