Fractional Moving Boundary Problems And Some Of Its Applications To Controlled Release System Of Drug

This paper is composed of four chapters, which are independent and correlative to one another. In chapter 1 i.e. introduction, the history, definitions, properties and applications of fractional calculus are introduced. In section§1.1 and§1.2, the development history and some definitions of the fractional calculus are introduced concisely. The definitions and the main properties of the Riemann-Liouville fractional integral operator (?) and differential operator (?) and the Caputo fractional derivative (?) are given. Some important properties of fractional integral and derivative operators are also discussed. In section§1.3, the definitions and some important formulae of the generalized Mittag-Leffler function E_α,β(z), the Wright function W_ρ,β and the generalized Wright function W_{(μ,a),(v,b)} are given. In section§1.4, the definition, series expression, asymptotic behavior and some basic properties of H-Fox function (?) are given. The special cases of the Fox function are discussed, such as the generalized Mittag-Leffler function E_α,β(z) and H_1,2^1,1(z). H-Fox function is a powerful tool for the solving of the fractional differential equations. In section§1.5, the developments and applications of fractional calculus in various fields are discussed, respectively. Section§1.6 gives a short introduction about the moving boundary problems and some of its application in drug release devices. This chapter is the basis for the following chapters of this thesis.In the following chapters, different models of a solute release from a planar polymer matrix are studied. In section§2.2, we give a detailed introduction about the mathematic model of the problem. We use the space-time fractional diffusion equation as the governing equation. Using a generalized flux equationand assuming a perfect sink, we obtain the following boundary conditions:andIn section§2.3, the solution of the model in form of Fox-H function is obtained with the help of Laplace and Fourier transforms. The concentration of the solute in the matrix isThe diffusion front S(t) can be written asThe constants q and p can be determined using the following equationsandA discussion is given in section§2.4, we can see that some results obtained previously are special cases of the model in this chapter. In chapter 3, we use the Riemman-Liouville and the Caputo fractional derivatives as the space fractional derivative in the model. In section§3.2, a scale-invariant variable z=xt^-α/β and the function of the diffusion front S(t) = pt^α/β are obtained by the Lie group method. The governing equation reduces to a fractional ordinary equationIn section§3.3, the solutions to the equations respect to the Riemman-Liouville and the Caputo fractional derivatives as the space fractional derivative are f(z) = (?) and f(z) = (?) correspondingly. C₁,C₂ and p can be decided byandIn the processing of the proof of our results, a alter form of the Caputo-type modification of Erderlyi-Kober fractional derivative operatorand the series expansion of Fox-H function are used. In section§3.4, the values of p in different cases are listed and we can see that the diffusion process described by the Caputo derivative is much faster than the one by the Riemann-Liouville derivative. The values of p in some cases are also shown in form of figures. In chapter 4, Homotopy perturbation method is successfully extended to solve time-fractional diffusion equation with a moving boundary condition and an approximate solution is obtained. The comparison with the exact solution shows that the approximate solution is sufficiently accurate for practical application in most cases. In section§3.2, the Homotopy perturbation method was introduced. By introducing a parameter p∈[0,1], we can construct the following homotopy:orand assume thatIn order to get the explicit form of p, the technique we used here is expanding the boundary conditions in its Taloy's seriesEquating the terms with identical powers of p, we can obtain a series of equations which are easier to solve. By pen-and-paper calculating, we can obtain the first order approximate solution written aswhere Introducing the fractional releasewe can give a comparison between the approximate with the exact solutions. Through the comparisons by the table and the figures. We can see that this approximate solution is concise and has a good degree of accuracy.