The Study Of Solutions Of Some Classes Of Initial Value Problem Of Fractional Differential Equations

Fractional calculus is the extension of the standard calculus with integer order, which researches on the theory and application of the differential and integral nonstandard operators of arbitrary order. It is an important branch of mathematical analysis. With the emergence of many fractal dimension facts in the nature and science, fractional calculus theory and fractional differential equations have got more confirmations of mathematicians and attracted more attention.In recent years, the theory of fractional calculus and fractional differential equations have got improvement and development, and been widely used in many fields. Hence, the study of the fractional calculus and fractional differential equations is practical and of great significance. Especially the fractional differential equation which is abstracted from practical problems becomes a research hotspot for many mathematicians.In this paper, we study the initial value problem of some classes of fractional differential equations. Some sufficient conditions on existence and uniqueness are established, which enriches the theory for the solution of initial value problem of fractional differential equations. The main work of this paper includes the following five parts:In the first chapter of introduction, we introduce the history and research background of fractional calculus and fractional differential equations, and briefly introduce some important results on the solution of existence and uniqueness of initial value problem of fractional differential equations.In the second chapter, we mainly discuss the simple types of fractional differential equations of weighted initial value problem. Using Schauder fixed point theorem, the nonlinear alternative of Leray-Schauder, upper and lower solution method, monotone iterative technique and corollaries of Banach contraction principle, we obtain some conditions on existence and uniqueness of solutions or positive solutions. In the third chapter, we mainly study two types of initial value problem of multi-order fractional differential equations. Using the nonlinear alternative of Leray-Schauder, the existence of positive solutions for multi-order fractional differential equation of weighted initial value problem is established. By means of upper and lower solution method, monotone iterative technique, Guo-Krasnosels'kii fixed point theorem, Schauder fixed point theorem and Banach contraction principle, several results on existence, uniqueness and multiplicity of solutions for initial value problem of multi-order fractional differential equations are given. And we present some examples to illustrate the main results.In the fourth chapter, we mainly study initial value problem of two types of the coupled system of fractional differential equations. Using Schauder fixed point theorem, the existence of solutions of a coupled system of fractional differential equations of weighted initial value problem is discussed. By means of Schauder fixed point theorem and Banach contraction principle, the existence and uniqueness of solutions of other coupled system of multi-order fractional differential equations are studied. And we give some examples to verify the main results.In the fifth chapter, we summarize the main work in this paper and point out the innovations of our work. Finally, we discuss the research of initial value problem of the fractional differential equation in the future.