New exact solutions of time conformable fractional Klein Kramer equation

Abstract

The Klein Kramer equation (KKE) stands for the probability distribution function (PDF) that describes the diffusion of particles subjected an external force in the presence of friction. It is applicable in statistical and stochastic treatments of chemical dynamics, in particular the diffusive description of chemical reactions. Here, we are concerned with finding the exact solutions of the conformable fractional derivative (CFD) KKE. An approach is presented to transform linear partial differential equations (PDEs) to a set of first order PDEs. On the other hand the CFD is shown to be reduced to the classical one’s by using similarity transformations. Here, the objective of this work is to find the exact solutions of CFD-KKE. To this issue, the approach presented is applied. The solutions are found by implementing extended unified method. It is found that, the integrability condition is that the external force is constant. The numerical results of the solutions are calculated and the are shown graphically. Calculations are carried by MATHEMATICA 12.