The Applications Of Fractional Calculus In Anomalous Diffusion Equations With Infinite Fractal Media

This paper is composed of three chapters, which are independent and cor-relative to one another. In chapter l,i.e. prior knowledge, the definition and history of fractional calculus are introduced. In section (?)1.1, the development history and recent applications of fractional calculus are 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 four special functions-Bessel function, Mattag-Leffler function, H-fox function and generalized H-fox function are discussed. In sec-tion (?)1.3, several integral transforms are discussed.In section (?)1.3, the Laplace transform and the infinite Hankel transform of fractional calculus are discussed. This chapter is the basis for the following chapters of this thesis.In chapter 2, we put Fick's law on fractal media in to the constitutive continuous equation on fractal Considering the existence orgin item, we can get a fractional reaction diffusion equation with a fractional oscillator in infinite fractal medium. The initial and boundary conditions of this equation are By applying Laplace transform, infinite Hankel transform of two H-fox func-tions and the theory of generalized H-fox function, we get the exact solution of the model in the form of generalized H-fox functionIn chapter 3, we consider more extensive initial conditions We get the solution of this problem At the end of this chapter, we discuss the solution in some special situations and compare it with the solution in chapter 2.