分数阶微积分在量子力学和非牛顿流体力学研究中的某些应用
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摘要
本论文由彼此相关的而又独立的五章组成。第一章为序言，简要介绍了本文所需的数学工具，即分数阶微积分理论和某些特殊函数。在§1．1节中，简要介绍了分数阶微积分的发展历史及其最近的应用，分别给出了Riemann-Liouville(R-L)分数阶积分算子_0D_t~(-β)、分数阶微分算子_0D_t~β(0＜Reβ＜1)、Caputo分数阶导数D_*~α和Riesz／Weyl分数阶导数_(-∞)D_x~μ(μ＞0)的定义和重要性质，并讨论了分数阶积分和微分算子的Laplace变换。在§1．2节中，简单介绍了广义Mittag-Leffler函数E_(α，β)(z)的定义及其某些重要公式。在§1．3节中，给出了H-Fox函数的定义、级数表达式及其基本性质，并讨论了H-Fox函数的特例及其Fourier正弦和余弦积分变换。Fox函数是求解分数阶微分方程的有力工具。本章是后面各章的基础。
第二章讨论了分数阶微积分在量子力学中的应用，主要研究了一维空间分数阶Schr(?)dinger方程的一些应用。首先，§2．2节给出了自由粒子满足的分数阶Schr(?)dinger方程
并利用积分变换和H-Fox函数及其性质求得自由粒子的波函数：
§2．3节求解了如下的无限深方形势阱
中粒子的波函数和能级分别为：
§2．4节讨论并求得了粒子贯穿如下方形势垒
的贯穿系数和反射系数分别为：
还讨论了具有能量E的粒子通过方形势阱的情形。第二章的最后我们讨论了量子散射问题中与分数阶Schr(?)dinger方程
等价的广义的Lippmann-Schwinger积分方程
并同样利用H-Fox函数及其性质确定了其Green函数的形式：
在第三章中，我们对比广义Oldroyd-B流体的本构关系：引入了修正的现象逻辑学的分数阶导数描述的Darcy定律：
在描述粘弹性流体的分数阶Darcy定律的基础上，研究了带分数阶导数的广义Oldroyd-B流体在多孔介质中的涡流运动模型：
我们利用分数阶导数的Laplace变换和Hankel变换，以及广义的Mittag-Leffler函数，分别给出了涡流速度场和温度场的精确解：
特别得，当η=0时，(17)式简化为广义Oldroyd-B流体在非多孔介质中的涡流速度场的解；而当α=1，β=1时，(17)式化为经典Oldroyd-B流体通过多孔介质的涡流速度场的解。当α=0，λ→0，η=0时，我们便得到广义二阶流体的涡流速度场的解，该结果与沈等人的结果一致，并包含了过去的经典结果作为其特例。
第四章在描述粘弹性流体的分数阶Darcy定律的基础上，研究了广义Oldroyd-B流体在多孔介质中由于平板的突然启动而引起的流动：即Stokes第一问题。运用Laplace变换和Fox函数的性质，给出了该问题的速度精确解：
特别地，当η=0时，上述结果就退化为非多孔介质中的广义Oldroyd-B流体的Stokes第一问题的解；当α=1，β=1可得到多孔介质中整数阶Oldroyd-B流体的相关结果；当α=0，λ=0时，可得到多孔介质中广义二阶流体的相关结果；当β=0，λ_t=0时，得到多孔介质中广义二阶流体的解。
第五章中讨论了带分数阶导数的广义Oldroyd-B流体的非定常Couette流模型：
应用Laplace变换和Weber变换，以及广义Mittag-Leffler函数，我们得到了该问题的精确解：
其中A(λ_i，t)=L~(-1)
特别地，当α=β=1时，该模型简化为经典Oldroyd-B流体的非定常Couette流模型；当β=0，λ_t→0时，结果即为广义Maxwell流体的非定常Couette流的解；当α=0，λ→0时，结果即为广义二阶流体的非定常Couette流的解；当α=0，λ=0，β=0，λ_t=0时，该结果简化为Newtonian粘性流体的非定常Couette流的经典解。
This paper is composed of five chapters, which are independent and correlative to one another. In chapterl, i.e. prologue, the fractional calculus and some special functions are introduced. They are the basic math tools needed in this paper. In section§1.1, the fractional calculus and its development history, current status are introduced concisely, the definitions and the main properties of the Riemann-Liouville fractional integral operator _oD_t~(-β), differential operator_oD_t~(-β)(00) are given. Andthe Laplace transforms of fractional integral and derivative operators are discussed. In section§1.2, the definitions and some important formulae of the generalizedMittag-Leffler function E_(α,β)(z) are given. In section§1.3, the definition of seriesexpression and some basic properties of H-Fox function are outlined. The special cases of Fox function and its Fourier sine and cosine transforms are discussed. H-Fox function is a powerful tool for the solving of the fractional differential equations. This chapter is the basis for the remainder.
In chapter 2, we investigate some applications of fractional calculus in quantum mechanics. Some physical applications of the space fractional Schrodinger equation are considered. In§2.2, using the properties of H-Fox function, we solve the time-dependent fractional Schrodinger equation for a free particle and give the wave function of it
In §2.3 the energy levels and the normalized wave functions of a particle in an infinite square potential well
are given as follows
respectively. In §2.4 according to the time-independent fractional Schrodinger equation we consider the motion of a particle from a rectangular potential wall
and give the reflection coefficient
and the transmission coefficient
When the particle is passing over a rectangular potential well instead of a potential
barrier, the above theory still holds, in which . In the last section, the fractional time-independent Schrodinger
equationand the equivalent Lippmann-Scheinger functionare considered, and the Green's function of it is given
In chapter 3, by analogy with generalized Oldroyd-B constitutive relationshipwe introduce a modified phenomenological Darcy's law with fractional derivative as followsBased on a modified Darcy's law for a viscoelastic fluid, the circular motion of a generalized Oldroyd-B fluid with fractional derivative model through a porous mediumis investigated. Using the discrete Laplace transform of the sequential fractional derivatives and Hankel transform, exact solutions of velocity and temperature fields are obtained in terms of generalized Mittag-Leffler function In special, whenη= 0, the above result can be simplified to the solution of vortex velocity field for a clear generalized Oldroyd-B fluid (nonporous case), and to be the solution for ordinary Oldroyd-B fluid through porous medium, whenα=1,β=1.Whenα=0,λ→0,η= 0, the solution of vortex velocity field for generalizedsecond grade fluid can be obtained, which is the same as the result obtaied by Shen and includes some past classical results as special.
In chapter 4, the fractional calculus approach has also been taken into account in the Darcy's law and the constitutive relationship of fluid model. Based on a modified Darcy's law for a viscoelastic fluid, Stokes' first problem is used to solve a generalized Oldroyd-B fluid problem in a porous half spaceBy using the Laplace transform method, an exact solution of Stokes' first problem in the porous half space is obtained as follows As special cases, we can obtain the Stokes' first problems for the generalized second grade fluid, for the fractional Maxwell fluid and for the ordinary Oldroyd-B fluid in porous medium, respectively.
In chapter 5, the unsteady Couette flow of generalized Oldroyd-B fluid with fractional derivative is studied. The dimensionless equations are:By using the Laplace transform, Weber transform and the generalized Mittag-Leffler function, we obtain the exact solutionSome previous results, such as the solution of unsteady Couette flow of generalized Maxwell fluid and of generalized second grade fluid, and the classical result of Newtonian viscous fluid all can be regarded as special cases of this solution.
第二章讨论了分数阶微积分在量子力学中的应用，主要研究了一维空间分数阶Schr(?)dinger方程的一些应用。首先，§2．2节给出了自由粒子满足的分数阶Schr(?)dinger方程
并利用积分变换和H-Fox函数及其性质求得自由粒子的波函数：
§2．3节求解了如下的无限深方形势阱
中粒子的波函数和能级分别为：
§2．4节讨论并求得了粒子贯穿如下方形势垒
的贯穿系数和反射系数分别为：
还讨论了具有能量E的粒子通过方形势阱的情形。第二章的最后我们讨论了量子散射问题中与分数阶Schr(?)dinger方程
等价的广义的Lippmann-Schwinger积分方程
并同样利用H-Fox函数及其性质确定了其Green函数的形式：
在第三章中，我们对比广义Oldroyd-B流体的本构关系：引入了修正的现象逻辑学的分数阶导数描述的Darcy定律：
在描述粘弹性流体的分数阶Darcy定律的基础上，研究了带分数阶导数的广义Oldroyd-B流体在多孔介质中的涡流运动模型：
我们利用分数阶导数的Laplace变换和Hankel变换，以及广义的Mittag-Leffler函数，分别给出了涡流速度场和温度场的精确解：
特别得，当η=0时，(17)式简化为广义Oldroyd-B流体在非多孔介质中的涡流速度场的解；而当α=1，β=1时，(17)式化为经典Oldroyd-B流体通过多孔介质的涡流速度场的解。当α=0，λ→0，η=0时，我们便得到广义二阶流体的涡流速度场的解，该结果与沈等人的结果一致，并包含了过去的经典结果作为其特例。
第四章在描述粘弹性流体的分数阶Darcy定律的基础上，研究了广义Oldroyd-B流体在多孔介质中由于平板的突然启动而引起的流动：即Stokes第一问题。运用Laplace变换和Fox函数的性质，给出了该问题的速度精确解：
特别地，当η=0时，上述结果就退化为非多孔介质中的广义Oldroyd-B流体的Stokes第一问题的解；当α=1，β=1可得到多孔介质中整数阶Oldroyd-B流体的相关结果；当α=0，λ=0时，可得到多孔介质中广义二阶流体的相关结果；当β=0，λ_t=0时，得到多孔介质中广义二阶流体的解。
第五章中讨论了带分数阶导数的广义Oldroyd-B流体的非定常Couette流模型：
应用Laplace变换和Weber变换，以及广义Mittag-Leffler函数，我们得到了该问题的精确解：
其中A(λ_i，t)=L~(-1)
特别地，当α=β=1时，该模型简化为经典Oldroyd-B流体的非定常Couette流模型；当β=0，λ_t→0时，结果即为广义Maxwell流体的非定常Couette流的解；当α=0，λ→0时，结果即为广义二阶流体的非定常Couette流的解；当α=0，λ=0，β=0，λ_t=0时，该结果简化为Newtonian粘性流体的非定常Couette流的经典解。
This paper is composed of five chapters, which are independent and correlative to one another. In chapterl, i.e. prologue, the fractional calculus and some special functions are introduced. They are the basic math tools needed in this paper. In section§1.1, the fractional calculus and its development history, current status are introduced concisely, the definitions and the main properties of the Riemann-Liouville fractional integral operator _oD_t~(-β), differential operator_oD_t~(-β)(0
In chapter 2, we investigate some applications of fractional calculus in quantum mechanics. Some physical applications of the space fractional Schrodinger equation are considered. In§2.2, using the properties of H-Fox function, we solve the time-dependent fractional Schrodinger equation for a free particle and give the wave function of it
In §2.3 the energy levels and the normalized wave functions of a particle in an infinite square potential well
are given as follows
respectively. In §2.4 according to the time-independent fractional Schrodinger equation we consider the motion of a particle from a rectangular potential wall
and give the reflection coefficient
and the transmission coefficient
When the particle is passing over a rectangular potential well instead of a potential
barrier, the above theory still holds, in which . In the last section, the fractional time-independent Schrodinger
equationand the equivalent Lippmann-Scheinger functionare considered, and the Green's function of it is given
In chapter 3, by analogy with generalized Oldroyd-B constitutive relationshipwe introduce a modified phenomenological Darcy's law with fractional derivative as followsBased on a modified Darcy's law for a viscoelastic fluid, the circular motion of a generalized Oldroyd-B fluid with fractional derivative model through a porous mediumis investigated. Using the discrete Laplace transform of the sequential fractional derivatives and Hankel transform, exact solutions of velocity and temperature fields are obtained in terms of generalized Mittag-Leffler function In special, whenη= 0, the above result can be simplified to the solution of vortex velocity field for a clear generalized Oldroyd-B fluid (nonporous case), and to be the solution for ordinary Oldroyd-B fluid through porous medium, whenα=1,β=1.Whenα=0,λ→0,η= 0, the solution of vortex velocity field for generalizedsecond grade fluid can be obtained, which is the same as the result obtaied by Shen and includes some past classical results as special.
In chapter 4, the fractional calculus approach has also been taken into account in the Darcy's law and the constitutive relationship of fluid model. Based on a modified Darcy's law for a viscoelastic fluid, Stokes' first problem is used to solve a generalized Oldroyd-B fluid problem in a porous half spaceBy using the Laplace transform method, an exact solution of Stokes' first problem in the porous half space is obtained as follows As special cases, we can obtain the Stokes' first problems for the generalized second grade fluid, for the fractional Maxwell fluid and for the ordinary Oldroyd-B fluid in porous medium, respectively.
In chapter 5, the unsteady Couette flow of generalized Oldroyd-B fluid with fractional derivative is studied. The dimensionless equations are:By using the Laplace transform, Weber transform and the generalized Mittag-Leffler function, we obtain the exact solutionSome previous results, such as the solution of unsteady Couette flow of generalized Maxwell fluid and of generalized second grade fluid, and the classical result of Newtonian viscous fluid all can be regarded as special cases of this solution.
引文
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