Research On Numerical Algorithms For Fractional Impulsive Differential Equations And Control Problems

Fractional calculus was proposed more than 300 years ago,but in recent decades,it has gradually become a hot topic in the study of scholars.Because the system described by fractional differential equation can better reflect the dynamic characteristics,it is widely used in many fields and engineering problems.With the higher and more accurate requirements for control and modeling in the practical application of engineering,we have to study the fractional theory from the perspective of optimal control.Therefore,the optimal control problem which takes fractional differential equation as the equation of state has become a new research hotspot and a research direction with application value.This paper mainly studies the numerical algorithm of fractional order impulse differential equation and optimal control problem.The first work is to construct a 2order numerical scheme for impulsive fractional differential equation by referring to the modified Block-by-block method and the midpoint rectangle formula,and analyze the local truncation error of the scheme.The convergence and stability are analyzed strictly.Finally,a typical numerical example is given to verify the effectiveness of the numerical algorithm and the correctness of the theoretical analysis.The second work in this paper is based on the Multi-Quadric(MQ)quasi interpolation operator,constructing the Multi-Quadric quasi-interpolation operator of the fractional order integral equation and the MQ quasi-interpolation operator of the fractional order differential equation.Firstly,two operators approximating the Hadamard fractional order integral-differential equation are given,and their properties and order of convergence are verified.Second,we get γ(0<γ<1)the convergence order of Hadamard fractional integration scheme is 3,and the convergence order of differential scheme is 3-γ.Finally,the numerical results show that MQ pseudo-interpolation method is an effective tool for constructing numerical formats.The third work of this paper is based on the second work,using the method of optimization before discretization,the radial basis function method in space,the finite difference method in time for discretization,construct the numerical format of Caputo fractional partial differential equation optimal control problem,analyze the local truncation error.Finally,a numerical example is used to verify the correctness of the theory.