The Fractional Reproducing Kernel Collocation Method For Solving The Fractional Fokker-planck Equation

Fractional calculus is widely used in many fields,including materials and ma-chinery,signal processing and system identification,control and robotics.A large number of physical systems are modeled by fractional differential equations,es-pecially fractional partial differential equations,which have great applications in various fields such as electrochemistry,circuits,theoretical biology and quantum mechanics.Among them,various generalized forms of the Fokker-Planck equation are widely used in physics,chemistry,engineering and biology.In the study of d-iffusion processes in many highly non-homogeneous media,the traditional integer order Fokker-Planck equation may not be suitable,so a model based on fractional derivative is proposed.Nonlocality is one of main advantages of fractional calculus which can be widely used,which makes fractional models often better fit experimen-tal data than integer-order models.On the other hand,fractional calculus is also difficult to calculate.Therefore,the main research content of this paper is to find an efficient and accurate numerical method to solve the time-fractional Fokker-Planck equation.In order to solve the time-fractional Fokker-Planck equation,a binary integer-fractional reproducing kernel collocation method is proposed.Firstly,a new binary integer-fractional reproducing kernel space^（3,）is constructed and its reproducing kernel function is given.Then a finite-dimensional subspace is generated based on its reproducing kernel function,and the time-fractional Fokker-Planck equation is solved in this subspace.On the one hand,the existence and uniqueness of the weak solution of the original problem are proved theoretically by Riesz representation theorem.On the other hand,by constructing the approximate solution based on the binary regenerated kernel function,combining the collocation method and the reproducing kernel method,it is proved that the constructed approximate solution and its partial derivative can converge uniformly to the exact solution and its partial derivative.Finally,the high accuracy and stability of the proposed algorithm are verified by four examples with different emphases.The innovations of this paper include:firstly,the fractional order reproducing kernel space is introduced to construct the binary integer-fractional order repro-ducing kernel space.The good property of fractional reproducing kernel function simplifies the calculation of fractional derivative.Secondly,the binary reproducing kernel function is used as the basis to construct the solution space and approxi-mate solution,and the collocation method is combined with the reproducing ker-nel method,which makes the system of collocation equation more easily obtained.Meanwhile,the property of the collocation method is simple and easy to operate,and the property of the reproducing kernel function is used to ensure the good con-vergence.Thirdly,the proposed algorithm does not distinguish the time variable and the space variable.Compared with the traditional numerical method,it avoids the differential dispersion of the fractional derivative of time and effectively reduces the error.