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In the present article, an approach to find the exact solution of the fractional Fokker-Planck equation 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, reduction of the fractional derivatives to non autonomous ordinary derivative. Thus the fractional Fokker-Planck equation 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. It is found that high friction coefficient plays a significant role in lowering the standard deviation. Further, it is found the fractionality has stronger effect than fractality. It is worthy to mention that the mixture of Gaussian's is a powerful tool in machine learning. Further, when 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.

We investigated the analytical solution of fractional order K(m,n) type equation with variable coefficient which is an extended type of KdV equations into a genuinely nonlinear dispersion regime. By using the Lie symmetry analysis, we obtain the Lie point symmetries for this type of time-fractional partial differential equations (PDE). Also we present the corresponding reduced fractional differential equations (FDEs) corresponding to the time-fractional K(m,n) type equation., Comment: 12 pages

In this paper, an efficient technique is employed to study the modified Boussinesq and approximate long wave equations of the Caputo fractional time derivative, namely q-homotopy analysis transform method. These equations are playing a vital rule in describing the properties of shallow water waves through distinct dispersion relation. The convergence analysis and error analysis has been presented in the present investigation for the future scheme. We illustrate two examples to demonstrate the leverage and effectiveness of the proposed scheme, and the error analysis has been discussed to verify the accuracy. The numerical simulation has been conducted to ensure the exactness of the future technique. The obtained numerical and graphical results are divulge, the proposed scheme is computationally very accurate and straightforward to study and find the solution for fractional coupled nonlinear complex phenomena arised in science and technology., Comment: 23 Pages, 18 Figures

To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation. In this paper, we develop Laguerre spectral collocation methods for solving variable-order fractional initial value problems on the half line. Specifically, we derive three-term recurrence relations to efficiently calculate the variable-order fractional integrals and derivatives of the modified generalized Laguerre polynomials, which lead to the corresponding fractional differentiation matrices that will be used to construct the collocation methods. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation methods., Comment: 12 pages and 1 figure