Generalized Klein–Kramers equation: solution and application

A generalized Klein–Kramers equation based on the continuous time random walk model is investigated. The equation generalizes the ordinary and fractional Klein–Kramers equations. Analytic solutions for the probability density and first two moments (for the force-free case) are obtained, and their dynamic behaviors are investigated in detail. The model is used to describe the cell migration of two migrating transformed renal epithelial Madin–Darby canine kidney (MDCK-F) cell strains: wild-type (NHE+) and NHE-deficient (NHE−) cells. Our theoretical predictions are in good agreement with experimental work in the paper (Dieterich et al 2008 Proc. Nat. Acad. Sci. USA 105 459). (paper)