Higher Order Numerical Algorithms For Fractional Differential Equations And Optimal Control Problems

Fractional calculus is a branch that studies the properties of any order integral or derivative.Fractional differential equation is an extension of integer order.It is widely used in various fields.In many engineering problems,it better describes the system than integer order.In the production process,it often requires some performance index of the system to be optimal.We have to study and analyze from the perspective of optimal control,and for fractional differential systems,we also have the optimal control problem relative to,so the fractional optimal control problem has been widely concerned by more and more scholars.In the first part of this paper,a high-order numerical scheme for Caputo nonlinear fractional ordinary differential equation is constructed by using piecewise quadratic Lagrange interpolation method,and then the local truncation error of the high-order numerical scheme is analyzed.Then based on the local truncation error,get 3- convergence order.Its stability and convergence are strictly analyzed.Finally,the highorder numerical scheme is numerically simulated.Through the calculation of typical problems,the effectiveness of numerical algorithm and the correctness of theoretical analysis are verified.In the second part of this paper,based on the first part,the numerical scheme of the optimal control problem of Caputo fractional partial differential equation is constructed by using the method of optimization before discretization,and then the finite difference is discretized in time and space,and its local truncation error is strictly analyzed.Finally,a numerical example is given to verify the effectiveness of the numerical scheme.