Ulam Stability Of Differential Equations With Caputo Fractional Derivative Operator

Fractional calculus is an important part of mathematical research.Because of the nonlocality of fractional derivative operators,on which some models with genetic and memory characteristics can be established,fractional caculus has been widely used in biology,physics and many other scientific fields.In recent years,there is an increasing number of studies on the stability of equations and Ulam stability,which play an important role in numerical analysis and optimization of equations.This thesis focuses on the Ulam stability of the mixed-order differential equation and the Caputo-Katugampola fractional differential equation.The first chapter briefly explains the background and significance of fractional differential equation,and introduces the main content of this paper.In the second chapter,some basic definitions,symbols,related theorems and lemmas of fractional calculus are made clear,and the concept of Ulam type stability is given.The third chapter mainly focuses on the stability of UlamHyers,Ulam-Hyers-rassias and semi-Ulam-Hyers-rassias for the Mixed-order differential equation in weighted spaces.We obtain the expected results by using the Banach Fixedpoint theorem and Laplace tranform,and the validity of the conclusion is verified by numerical experiments.The mixed-order differential equation can represent the constitutive relation of the viscoelastic model,and can also describe the macroscopic model of ion electric diffusion in nerve cells when molecular diffusion is abnormal(II)secondary diffusion due to binding,crowding or trapping.In next chapter,we mainly study the existence and uniqueness of fractional differential equation solutions with Caputo-Katugampola.The existence of the solution of the equation is obtained by the contraction mapping principle,and the uniqueness of the solution is pro-ved by the Schauder and Wessinger Fixed-point theorem,and the solution of the equation is extended from local to global.Based on Chapter Three and Four,in Chapter Five,the UlamHyers-Mittag-Leffler and Ulam-Hyers-Rassias-Mittag-Leffler stability of fractional differential equation is proved by means of successive approximation and Gronwall inequality.Chapter 6,Summarizes the full text and looks forward to future work.