Different Kinds Of Exact Solutions Of Fractional Order Fokker-Planck Equation And Corresponding Dynamic Phenomena

Many problems of non-Newtonian mechanics in natural science and engineering can be summed up as solving the problem of time-fractional partial differential equation model to be studied.The mathematical models of these complex problems have profound physical background and rich theoretical connotation,and have a very wide range of applications in physics,chemistry,biology,economics and many other subjects.It is very important to develop the methods of solving these fractional partial differential equations and finding their exact solutions and explain their dynamic phenomena.In this work,the time-fractional Fokker-Planck equation with external potential field is studied by using the classical variable separation method and the properties of MittagLeffler function.Under different external potential fields such as linear potential,harmonic potential,logarithmic potential,exponential potential and quartic potential,we systematically study different kinds of exact solutions and their dynamic properties of the time-fractional Fokker-Planck equation,and draw three-dimensional and two-dimensional evolution graphs of some representative exact solutions.Through the evolution of these graphs and the analysis of the dynamic properties of the solutions,we further explain the different kinds of dynamic phenomena in the model.The research work in this paper is different from the numerical solutions of the time-fractional Fokker-Planck equation in previous references.The research work in this paper is mainly based on the exact solutions of the model and the study of dynamic properties and dynamic phenomena.Obviously,compared with the numerical solutions obtained in the existing references,the exact solutions obtained in this paper can not only describe the dynamic phenomenon of the model more accurately,but also systematically reveal the inherent operation law of the model.The innovation of this paper has three aspects: one is the innovation of solving method,that is,the method combining the separation of variables method and the Mittag-Leffler function is proposed.This method transforms the fractional partial differential equation into ordinary differential equation and simplifies the solving process;The second is the innovation in the form design of exact solutions,that is,the innovation in the structure hypothesis of solutions.A large number of new results are obtained under the new structure hypothesis of solutions.The third is the innovation in the study of the exact solution of the time fractional Fokker-Planck equation and its dynamic properties,which breaks the situation that previous scholars only study the numerical solution and approximate analytical solution of the time fractional Fokker-Planck equation.