The Solution Of Fractional Anomalous Diffusion Equation

In the present paper, Fractional nonlinear convection-diffusion equation in the absence of external force is given. Frist, we study the integer nonlinear convection-diffusion equation by considering the diffusion coefficient D{x) ∝ x^-θ and the external force, by applying the properties of the q exponential function and q logarithm function, analysis of the equation is derived. Second, we analyze a fractional nonlinear convection-diffusion equation with absorption, if the nonlinear parameters satisfy some conditions, a special solution can be obtained. And the asymptotic behaviors for the solution are also discussed. At the same time fractional reaction diffusion integral equations with absorption in finite fractal media are given. Analytical solution of the concentration distribution can be expressed as the Mittag-leffler function by means of Laplace transform and generalized finite Hankel transform. And the asymptotic behaviors for the solution are also given. It offers a method to solve the anomalous diffusion equation in disorded media.This paper is composed of four chapters, which are independent and correlative to one another. In chapter 1, i. e. prologue, the fractional calculus and its history, current status are introduced. It's the basic tool needed in this paper. In sections §1.1 §1.3, the development history and the recent applications of the fractional calculus are introduced concisely, the definitions and the main properties of the Riemann Liouville fractional integral operator 0D_t^-β(0 < Reβ < 1) and differential operator 0D_t^λ(0 < Reλ < 1) are given, and the Laplace transforms of fractional integration and differentiation are discussed. In section §1.4, the definition and basic properties of the generalized Mittag Leffler function E_α,β(z) are given. In section §1.5: generalized finite Hankel transform and inverse transform are given. This chapter is a basis for the following all charpters.In chapter 2, fractional nonlinear convection-diffusion equation in the absence of external force is given as follows:...