Ulam-hyers Stability For Fractional Differential Equations

Fractional calculus is an indispensable part of applied mathematics research.Due to the nonlocality property of fractional derivative operators,they can be used to describe some models with genetic and memory characteristics,and they have been widely used in many scientific fields such as physics and biology.Ulam stability plays a crucial role in numerical analysis,optimization and other fields.In the past two decades,many scholars have paid more and more attention to the concept of Ulam stability.In this paper,the existence,uniqueness and Ulam type stabilities of solutions of fractional differential equations are studied.First of all,the background and significance of this paper at home and abroad are introduced,as well as the basic knowledge and relevant lemma.Then,the analytical solution of a nonlinear Caputo fractional reaction-diffusionconvection problem is discussed by two methods in chapter 3.What’s more,chapter 4considered a class of Caputo mixed fractional order and integer order differential equations with integral boundary conditions.In terms of contraction mapping principle and Krasnosel’skiǐ fixed point theorem,this paper proposed the conditions of the existence and uniqueness of solutions.By means of the Banach fixed point theorem,the Ulam-Hyers-Rassias and Ulam-Hyers stability of the nonlinear fractional differential equations are investigated.Futhermore,chapter 5 introduced a class of implicit fractional differential equations involving ψ-Hilfer fractional derivative.We considered the existence and uniqueness of solutions to the nonlinear fractional differential equations by contraction mapping principle.Finally,the Ulam-Hyers-Rassias and Ulam-Hyers stability of the fractional differential equation are investigated in terms of the Banach fixed point theorem in the weighted space.Finally,an example is discussed to illustrate the main work in each section.