Finite Difference Method For The Nonlinear Differential Equation With Two Caputo Fractional Derivatives

In recent years,with the rapid development of fractional order nonlinear differential equations and their wide application in many scientific fields,fractional order nonlinear differential equations have attracted more and more scholars' attention.However,it is so hard to express the analytical solutions of fractional order nonlinear differential equations explicitly.Therefore,it is of great significance to further study the effective numerical methods for solving fractional order nonlinear differential equations.In this paper,we study the existence and uniqueness,Ulam-Hyers stability and two effective numerical methods for solving the initial value problem of nonlinear differential equation with two Caputo fractional derivatives.The main content of this paper includes:In the first part,we study the initial value problem of nonlinear differential equation with two Caputo fractional derivatives,and prove the existence and uniqueness,Ulam-Hyers stability of the equation.In the second part,the L1 difference algorithm for solving nonlinear differential equation with two Caputo fractional derivatives is constructed by applying the L1 interpolation to approximate the Caputo fractional derivatives.Then,the stability and convergence of the numerical method are proved.In the third part,the L1-2 difference algorithm for solving nonlinear differential equation with two Caputo fractional derivatives is constructed by using L1-2interpolation to approximate the Caputo fractional derivatives.Next,the stability and convergence of the numerical method are proved.Finally,the feasibility of the method is verified by numerical experiments.