The Applications Of Fractional Calculus In Quantum Mechanics And Anomalous Diffusion Equations

This paper is composed of three chapters, which are independent and correlative to one another. In chapter 1. prior knowledge, the definition and history of fractional calculusare introduced. In section§1.1, the development history and recent applications of fractional calculus arc introduced concisely, the definitions and main properties of the Riemann-liouville fractional operator and Caputo fractional operator are given. In section§1.2 the definitions and important formulae of two special functions-Mattag-Leffier function and H-Fox function are discussed. In section§1.3, several integral transforms are discussed.In section§1.4, the Fourier transform,tho Laplace transform,the finite Hankel transform of fractional calculus are discussed. This chapter is the basis for the following chapters of this thesis.In chapter 2, on the basis of preceding research of Schrodinger equation,we study the fractional Schrodinger equation with a doubleδ-potential barrier:Furthermore, we discuss the jump conditions of the equation The corresponding reflection coefficient and transmission coefficient of the particle are given. And we discuss the relations between fractional and classical Schr(?)dingcr equation,and we can find that the result we got is consistent with [29]. In the end, we obtain the solution to the space fractional Schrodinger equation with a doubleδ-potential barrierIn chapter 3. fractional reaction-diffusion differential equation with fractional oscillatorin finite fractal medium is establishedBy applying Laplace transform, the finite Hankel transform and their inverse transforms,we get the exact solution of the model in the form of generalized Mittag-Lefffler functionWe also discuss the solutions of two-dimensional space, three-dimensional space and the integral diffusion equation as some particular cases of this paper. Finally, we discuss the the equation with no fractional oscillatorand the exact solutions can be written as followThis is proved to be the same with [35].