Exact solutions of the fractional time-derivative Fokker-Planck equation: A novel approach

In the present article, an approach to find the exact solution of the fractional Fokker-Planck equation (FFPE) is presented. It is based on transforming it to a system of first-order partial differential equation via Hopf transformation, together with implementing the extended unified method. On the other hand, a theorem provides the reduction of the fractional derivatives to non-autonomous ordinary derivative is given. Thus, the FFPE is reduced to non-autonomous classical ones. Some explicit solutions of the classical, fractional time-derivative Fokker-Planck equation are obtained. It is shown that the solution of the Fokker-Planck equation is bi-Gaussian's, which was not found up to date. It is found that high friction coefficient plays a significant role in lowering the standard deviation. Further, it is found that the effect of the presence of the fractional derivative prevails that of the fractal derivative. Here, the most interesting result found is that mixed-Gaussian solution is obtained. It is worthy to mention that the mixture of Gaussian's is a powerful tool in machine learning and also in the distribution of loads in networks. Further, varying the order of the fractional time derivatives results to slight effects in the probability distribution function. Also, it is shown that the mean and mean square of the velocity vary slowly.