Anomalous Diffusion Models Involving Regularized General Fractional Derivatives with Sonin Kernels - Maryam Alkandari; Dimitri Loutchko; Yuri Luchko (Fractal and Fractional)
In this paper, we introduce a general fractional master equation involving regularized general fractional derivatives with Sonin kernels, and we discuss its physical characteristics and mathematical properties. First, we show that this master equation can be embedded into the framework of continuous time random walks, and we derive an explicit formula for the waiting time probability density function of the continuous time random walk model in form of a convolution series generated by the Sonin kernel associated with the kernel of the regularized general fractional derivative. Next, we derive a fractional diffusion equation involving regularized general fractional derivatives with Sonin kernels from the continuous time random walk model in the asymptotical sense of long times and large distances. Another important result presented in this paper is a concise formula for the mean squared displacement of the particles governed by this fractional diffusion equation. Finally, we discuss several mathematical aspects of the fractional diffusion equation involving regularized general fractional derivatives with Sonin kernels, including the non-negativity of its fundamental solution and the validity of an appropriately formulated maximum principle for its solutions on the bounded domains.