Models of Anomalous Diffusion and its analysis

Abstract

The anomalous diffusion is characterized by deviation from Gaussian statistics and the absence of a linear time dependency in the mean square displacement. This study investigates anomalous diffusion processes that exhibit a power-law growth in mean square displacement as time progresses.

The first stage is to create the model using random methods, that is, by using random walks. The following statement describes a continuous-time random walk model represented by a series of convolution-type integral equations depicting probability density functions. Fractional differential equations for time and space are derived from the master equations by choosing probability density functions with infinite first and/or second moments.

The obtained model equations are analyzed with respect to elementary boundary value problems in constrained fields. The main focus is on studying elementary boundary value problems related to the generalized fractional time diffusion equation, especially using the fractional Caputo derivative.

This equation applies a well-established maximum principle to stochastic partial differential equations of the elliptic and parabolic type (SPDEs). This concept is used to make preliminary estimates of the answer before using it to prove the uniqueness of the solution. To prove the existence of a solution, a clearly defined generalized solution is first generated using the spectroscopic method. Under certain additional circumstances, a comprehensive solution may be considered a solution in the traditional sense.