Characteristic Root Solution Of Fractional Differential Equations With Delays

In this paper,we mainly discuss the eigenvalue solution of fractional calculus.Considering the general solution formula with delay term under the Caputo type fractional calculus of exponential function,we study the eigenvalue solution of Riemann-Liouville type non-homogeneous fractional differential equation with delay term.The basis of this paper is fractional differential equation,which is the combination of fractional calculus and classical ordinary differential equation.It is a generalization of the eigenvalue method of ordinary differential equation.The structure of this paper is as follows:In the first chapter,the background knowledge points of fractional order and the current research situation are stated.A series of basic knowledge points of corresponding fractional order equation are introduced.The corresponding lemmas and theorems of eigenvalue solutions for ordinary differential equations are reviewed.Chapter 2 is the main part of this paper.This chapter mainly uses the general form of exponential function under Caputo derivative to generalize it to the case with time delay,and studies the general form of the general solution of the homogeneous equation containing this kind of function through the thought of classified discussion.At the end of this chapter,corresponding examples are given to illustrate it.In Chapter 3,based on the general solutions of homogeneous equations,the general solutions of Riemann-Liouville exponential functions with time delay are studied.The special solution of this kind of equation is deduced by the method used in ordinary differential equation,and according to the lemma,the solution of two cases is given,so that the form of the solution can be unified.