Some Results Of Fractional Differential Equations With Two Moving Boundaries In One Dimension And Its Applications

This paper is composed of three chapters, which are independent and correlative to each other.In chapter 1, some elementary knowledge, history and applications of fractional calculus are introduced. In section 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 also discussed. In section 2, the integral transforms are introduced. In section 3, the special functions, which play important role in fractional differential equations, are introduced. The definitions and some important formulae of the generalized Mittag-Leffler function E_α,β(z), Wright function and H-Fox function H_p,q^m,n(z) are given. Special functions are the powerful tools for solving of the fractional differential equations.In chapter 2, we set up a one dimension mathematical model of drug released from polymeric matrix that can be dissolved in the solvent using the tool of fractional calculus. Assuming that the matrix solvent can be dissolved, it will dissolve when the drug are dissolved in the solvent. Then there will be two moving boundaries: the dissolving boundary and diffusing boundary. It is called two moving boundaries problems. The appearance of the nonlinear terms caused by moving boundaries makes the problem difficult to solve, and there are very few exact solutions to moving boundary problem. In section 1, the history and recent research of moving boundary problem are introduced concisely. In section 2, we point out that the model describing the drug release from a polymeric matrix will be more accurate by using the fractional derivative operators. And we generalize the Fick's law and set up a mathematical model of drug release from a dissolvable matrix using fractional operator with the assumption of perfect sink condition. In section 3, the special two boundaries problem is present. The Fourier transform and Laplace transform and two-parameter perturbation method are used to solve the nonlinear problem when the matrix is dissolved slowly. Then we get the non-dimension approximate solution that is expressed by Wright function:θ(y,t;η,ε) =θ₀(y,t)+θ₁(y,t)η+θ₂(y,t)ε+ .... whereIn section 4, we analyze the approximate solution.Chapter 3 is a summary of this paper. We obtain the conclusion that the fractional calculus can be used in the moving boundary problem and get good effects.