Mean-squared displacements for normal and anomalous diffusion of grains

The problem of normal and anomalous space difiusion is formulated on the basis of the integral equations with various type of the probability transition functions for difiusion (PTD functions). For the cases of stationary and time-independent PTD functions the method of fractional differentiation is avoided to construct the correct probability distributions for arbitrary distances, what is important for applications to different stochastic problems. A new general integral equation for the particle distribution, which contains the time-dependent PTD function with one or, for more complicated physical situations, with two times, is formulated and discussed. On this basis fractional differentiation in time is also avoided and a wide class of time dependent PTD functions can be investigated. Calculations of the mean-squared displacements for the various cases are performed on the basis of formulated approach. The particular problems for the PTD functions, dependable from one and for two times, are solved.