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Published in:
2014 | OriginalPaper | Chapter
Two Models of Subdiffusion Processes: When Are They Similar?
Authors : T. Kosztołowicz, K. D. Lewandowska
Published in: Advances in Applied Mathematics
Publisher: Springer International Publishing
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Abstract
We study two models which describe subdiffusive processes. Subdiffusion is defined by the relation \(\left \langle (\Delta x)^{2}(t)\right \rangle = D_{\alpha }t^{\alpha }\), where \(\left \langle (\Delta x)^{2}(t)\right \rangle\) denotes the mean square displacement, α is a subdiffusion parameter which obeys 0 < α < 1 and D
α
is a subdiffusion coefficient. The first model consists of a nonlinear partial differential equation with derivatives of a natural order obtained from a Sharma–Mittal nonadditive entropy, whereas the second model is based on a linear partial differential equation with a fractional time derivative which is derived from the continuous time random walk formalism. We obtain the fundamental solutions for both models. Next, we assume two agreement conditions. According to the first one the fundamental solutions for both model fulfill the relation which defines subdiffusion. The second agreement condition presumes the equality of the first passage time distributions. On the basis of these agreement conditions we answer the question when the considered models are similar.