分数阶可动边界问题及其在药物控释系统中的某些应用
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摘要
本论文由彼此相关而又独立的四章构成。第一章为序言,简要介绍了本文所需的数学工具,也即分数阶微积分的发展历史、基本概念、性质及应用。在§1.1和§1.2节,简要的介绍了分数阶微积分的历史,给出了Riemann-Liouville型分数阶积分算子(?)、微分算子(?)和Caputo型分数阶算子(?)的定义及主要性质,并讨论了分数阶积分和微分算子的Laplace变换。在§1.3节中,给出了Mittag-Leffler函数、广义Mittag-Leffler函数E_(α,β)(z)、Wright函数W_(ρ,β)(z)和广义Wright函数W_((μ,a),(v,b))(z)的定义及其某些重要公式。在§1.4节中,给出了H-Fox函数(?)的定义、级数表达式、渐近性态及其基本性质,并讨论了H-Fox函数的特例,如广义Mittag-Leffler函数既E_(α,β)(z)和(?)。Fox-H函数是求解分数阶微分方程的有力工具。在§1.5节,从非牛顿流体、反常扩散等几个方面简要阐述了分数阶微积分理论在几个领域内的研究进展状况。§1.6节简要介绍了可动边界问题及其在药物控释系统中的某些应用.本章是以后各章的基础。
接下来的几章研究了溶质从高分子基质中溶出的不同模型。在§2.2节给出了上述问题的详细介绍,并且应用时间-空间分数阶扩散方程作为描述扩散的主控方程。再应用推广了的通量方程并假设一个完全汇条件,得到了如下的边界条件:和在§2.3节,应用Laplace和Fourier变换,得到了以Fox-H函数表述的上述方程的解,其中,基质中溶质浓度的表达式为溶蚀边界S(t)可以写作上面两式中的常数q和p可由和这两个方程确定。§2.4节对所得解做了讨论,可以看出,之前的一些结果是本文结果的特殊情况。
在第3章,我们应用Riemman-Liouville和Caputo型的分数阶微分算子作为模型中的空间分数阶算子。在§3.2节,通过Lie群分析的方法,得到了一个相似变量z=xt~(-α/β)和溶蚀边界的函数表达式S(t) = pt~(α/β)。相应的,主控方程变成了如下的分数阶常微分方程在§3.3节,给出了对应于Riemman-Liouville和Caputo型算子的方程组的解,它们分别是f(z)=(?)和f(z)=(?)。C_1,C_2和p通过两组方程和分别确定。在证明所得结果的过程中,应用了Caputo型的修正Erderlyi-Kober分数阶微分算子的等价形式和Fox-H函数的级数形式。在§3.4节,列出了常数p在不同情形下的的取值,通过比较可以看出,用Caputo型算子描述的快于用Riemann-Liouville型算子所描述的扩散过程。p的不同取值同时也通过图像的形式给予了展示。
在第4章,同伦摄动的方法被成功应用到求解带有一个可动边条件的时间分数阶扩散方程并且得到了一个近似解。精确解和近似解的比较显示,近似解在大多数情况下对实际应用来说足够精确。§3.2节引入了同伦摄动的方法,通过引进一个参数p∈[0,1],我们可以建立如下的方程:或者写作并且假设为了得到参数p的显式表达式,本章应用的技巧是将边界条件展成其相应的泰勒级数形式:通过比较参数p的相同幂次,可以得到一系列的容易求解的方程组。通过简单的计算,我们可以得到问题的一阶近似解其中引进释放分数我们可以比较近似解和精确解。表格和图像的比较显示,近似解更加简洁,并且具有很好的精确度。
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. C_1,C_2 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.
接下来的几章研究了溶质从高分子基质中溶出的不同模型。在§2.2节给出了上述问题的详细介绍,并且应用时间-空间分数阶扩散方程作为描述扩散的主控方程。再应用推广了的通量方程并假设一个完全汇条件,得到了如下的边界条件:和在§2.3节,应用Laplace和Fourier变换,得到了以Fox-H函数表述的上述方程的解,其中,基质中溶质浓度的表达式为溶蚀边界S(t)可以写作上面两式中的常数q和p可由和这两个方程确定。§2.4节对所得解做了讨论,可以看出,之前的一些结果是本文结果的特殊情况。
在第3章,我们应用Riemman-Liouville和Caputo型的分数阶微分算子作为模型中的空间分数阶算子。在§3.2节,通过Lie群分析的方法,得到了一个相似变量z=xt~(-α/β)和溶蚀边界的函数表达式S(t) = pt~(α/β)。相应的,主控方程变成了如下的分数阶常微分方程在§3.3节,给出了对应于Riemman-Liouville和Caputo型算子的方程组的解,它们分别是f(z)=(?)和f(z)=(?)。C_1,C_2和p通过两组方程和分别确定。在证明所得结果的过程中,应用了Caputo型的修正Erderlyi-Kober分数阶微分算子的等价形式和Fox-H函数的级数形式。在§3.4节,列出了常数p在不同情形下的的取值,通过比较可以看出,用Caputo型算子描述的快于用Riemann-Liouville型算子所描述的扩散过程。p的不同取值同时也通过图像的形式给予了展示。
在第4章,同伦摄动的方法被成功应用到求解带有一个可动边条件的时间分数阶扩散方程并且得到了一个近似解。精确解和近似解的比较显示,近似解在大多数情况下对实际应用来说足够精确。§3.2节引入了同伦摄动的方法,通过引进一个参数p∈[0,1],我们可以建立如下的方程:或者写作并且假设为了得到参数p的显式表达式,本章应用的技巧是将边界条件展成其相应的泰勒级数形式:通过比较参数p的相同幂次,可以得到一系列的容易求解的方程组。通过简单的计算,我们可以得到问题的一阶近似解其中引进释放分数我们可以比较近似解和精确解。表格和图像的比较显示,近似解更加简洁,并且具有很好的精确度。
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. C_1,C_2 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.
引文
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