Fractional diffusion equation on fractals; Self-similar stationary solutions in a force field derived from a logarithmic potential
Consider the fractional diffusion equation in the form \[ {{\partial^{1/2} C} \over {\partial^{1/2} t}}= D^{1/2} \Biggl( {{\partial C} \over r} + {{n-1} \over r} C\Biggr) \qquad \text{as } r\to\infty, \tag{\(*\)} \] where \[ {{d^ a C} \over {dt^ 2}}= {1\over {\Gamma(1- a)}} {d\over {dt}} \int_ 0^ t {{C(r,\tau)} \over {(t- \tau)^ a}} d\tau, \qquad \Gamma(x)= \int_ 0^ \infty y^{x-1} e^{-y} dy, \] is the fractional time derivative of \(C(r,t)\) of order \(a\in (0,1]\). The author generalizes the fractional diffusion equation \((*)\) for Euclidean diffusion in force fields. The heuristic argument of Giona and Roman is used for generalizing the fractional diffusion equation in force fields to the fractal case. In this paper the author solves the problem of fluctuating fractal concentration profiles by using the theory of random point processes [see \textit{N. G. Van Kampen}, Stochastic processes in physics and chemistry (1981; Zbl 0511.60038); \textit{D. L. Snyder}, Random point processes (1975; Zbl 0385.60052); \textit{D. R. Cox} and \textit{V. Isham}, Point processes (1980; Zbl 0441.60053); \textit{S. K. Srinivasan}, Stochastic point processes and their applications (1974; Zbl 0287.60058)].
scientific article; zbMATH DE number 562531