Fast Higher Order Alogrithms Research For Fractional Differential Equations With Capoto-Fabrizio Derivatives

Fractional calculus as a generalization of integer calculus plays an important role in the fields of physics?chemistry?biology and fluid mechanics.Therefore,More and more mathematicians and physicists begin to have a strong interest in the study of fractional calculus,which also promotes the continuous improvement and development of fractional calculus theory to a certain extent.This paper mainly focuses on the solution of fractional ordinary differential equation with Caputo-Fabrizio derivative and the solution of Fokker-Planck equa-tion with Caputo-Fabrizio fractional derivative.Firstly,The paper based on the L2 scheme and simple recursive relation raise a fast and high-order numerical method to solve fractional ordinary differential equations with Caputo-Fabrizio derivatives.This algorithm greatly reduces the storage capacity and the total computing cost.In addition,the feasibility of the algorithm?the error estimation and the stability analysis of the fast scheme are also analyzed.Finally,the effectiveness and applica-bility of the proposed method are proved by numerical examples.Secondly,the Fokker-Planck equation with Caputo-Fabrizio fractional deriva-tive is effectively solved by using the fourth order scheme.The method is based on the finite difference method in time and the Legendre spectrum method in space.The error estimation and stability of the algorithm are established strictly.Finally,the correctness of the theory is verified by numerical experiment.