Solutions of the time fractional partial equations and applications

分数阶微积分是专门研究任意阶积分和微分的数学性质及其应用的领域，是传统的整数阶微积分的推广，分数阶微分方程是含有非整数阶导数的方程。近几十年里，研究者们发现分数阶微分方程非常适合用来描述现实生活中具有记忆和遗传特性的问题，如：分形和多孔介质中的弥散，电容理论，电解化学，半导体物理、湍流、凝聚态物理，粘弹性系统，生物数学及统计力学等等，因此研究这类方程的性质和数值解法有现实的理论和应用意义。 本文主要讨论一类时间分数阶空间二阶偏微分方程，讨论其解析解，数值解。 第一章，给出本论文的研究背景和意义，总结了前人所做的工作，并叙述分数阶微积分的概念和分数阶微积分一些基本定义和性质，详列本论文的研究...Fractional calculus is a branch of studying the property of any order integral or derivative.Fractional order differential equation is the equation containingthe non-integer order derivative, raising from the standard differential equations by replacing the integer-order derivatives with fractional-order derivatives. Its application is very broad, many researchers find that the fractional differen...学位：理学博士院系专业：数学科学学院信息与计算数学系_计算数学学号：B2005140300

非定常不可压粘性/ 无粘性耦合方程

给出了数值求解初始变量不可压Navier2Stokes/ Euler 耦合方程的一种分步块LU 分解方 法。与传统的时间分裂法不同,该法无需压力中介边条件,从而避免了传统时间分裂法要求 的复杂的压力中介边条件逼近。分步块LU 分解方法可看做经典的Uzawa 算法的改进,后者 曾被成功应用于不可压Navier2Stokes/ Euler 耦合方程的求解。但本文显示分步块LU 分解法 比经典的Uzawa 方法更经济。分析显示该法具有良好的稳定性和高精度,数值结果支持这一 理论分析。国家自然科学基金19801028 项目,科技部“中法先进研究计划”PRAS199203 项

教学原则在《高等数学》教学中的体现

本文通过对《高等数学》中的有关教学原则的理解 ,探讨“科学性与思想性相结合原则” ,“知识积累与智能发展相结合原则” ,“理论联系实际原则”等教学原则在《高等数学》教学中的运用及其特有的体现形

A fractional step method for the time dependent incompressible Navier2Stokes/ Euler coupled equations

摘要:给出了数值求解初始变量不可压Navier2Stokes/ Euler 耦合方程的一种分步块LU 分解方 法。与传统的时间分裂法不同,该法无需压力中介边条件,从而避免了传统时间分裂法要求 的复杂的压力中介边条件逼近。分步块LU 分解方法可看做经典的Uzawa 算法的改进,后者 曾被成功应用于不可压Navier2Stokes/ Euler 耦合方程的求解。但本文显示分步块LU 分解法 比经典的Uzawa 方法更经济。分析显示该法具有良好的稳定性和高精度,数值结果支持这一 理论分析。 　Abstract : A fractional step method for solving the incompressible Navier2Stokes/ Euler coupled equations in primitive variables is analyzed as a block LU decomposition. In this formulation , no in2 termediate boundary conditions for the velocity and the pressure is required , in contrast to the tradi2 tional time splitting method. In addition , the fractional step method can be reviewed as an improve2 ment of the classical Uzawa algorithm that has been proven to be useful in the calculation of the in2compressible Navier2Stokes/ Euler coupled equations. The fractional step method is showed to be cheaper than the classical Uzawa algorithm. The stability and the accuracy analysis are also given. Numerical results confirm the theoretical analysis. Simulations of the flow past the cylinder are car2 ried out in order to demonstrate the potential applications of the Navier2Stokes/ Euler coupled solver.国家自然科学基金19801028 项目,科技部“中法先进研究计划”PRAS199203 项目

A fractional step method for the time dependent incompressible Navier-Stokes/Euler coupled equations

给出了数值求解初始变量不可压Navier Stokes/Euler耦合方程的一种分步块LU分解方法。与传统的时间分裂法不同,该法无需压力中介边条件,从而避免了传统时间分裂法要求的复杂的压力中介边条件逼近。分步块LU分解方法可看做经典的Uzawa算法的改进,后者曾被成功应用于不可压Navier Stokes/Euler耦合方程的求解。但本文显示分步块LU分解法比经典的Uzawa方法更经济。分析显示该法具有良好的稳定性和高精度,数值结果支持这一理论分析。A fractional step method for solving the incompressible NavierStokes/Euler coupled equations in primitive variables is analyzed as a block LU decomposition. In this formulation, no intermediate boundary conditions for the velocity and the pressure is required, in contrast to the traditional time splitting method. In addition, the fractional step method can be reviewed as an improvement of the classical Uzawa algorithm that has been proven to be useful in the calculation of the incompressible NavierStokes/Euler coupled equations. The fractional step method is showed to be cheaper than the classical Uzawa algorithm. The stability and the accuracy analysis are also given. Numerical results confirm the theoretical analysis. Simulations of the flow past the cylinder are carried out in order to demonstrate the potential applications of the NavierStokes/Euler coupled solver.国家自然科学基金19801028项目;; 科技部"中法先进研究计划"PRAS199 03项目

Analytical Solution for the Non-homogeneous Anomalous Sub-diffusion Equation

扩散、对流-扩散和Fokker-Planck型的分数阶动力方程为描述在复杂系统中由反常扩散控制的传送动力学提供实用的近似.利用分离变量方法和Laplace变换分别导出在Dirichlet、Neumann和Robin边界条件下的非齐次反常次扩散方程的解析解.这个技巧可以推广到解其它类型的反常扩散方程.Fractional kinetic equations of the diffusion,diffusion-advection,and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion.The theoretical justification for the fractional diffusion equation,together with the abundance of physical and biological experiments demonstrating the prevalence of anomalous sub-diffusion(ASub-DE),has led to an intensive effort in recent years to find accurate and stable methods of solution that are also straightforward to implement.However,effective methods for the ASub-DE are still in their infancy.In this paper,using separation of variable methods and Laplace transform,the analytical solutions of a non-homogeneous ASub-DE with Dirichlet,Neumann and Robin boundary conditions are derived,respectively.These techniques can be applied to solve other kinds of anomalous diffusion.福建省自然科学基金(S0750017)资

Poiseuille-Benard流的出口边界条件及其谱元法计算

研究二维矩形管道中底部加热的不可压缩poiseuille-benard流的谱元法数值计算问题. 讨论各种不同的出口边界条件的处理及其对谱元法数值模拟的影响. 通过干扰区 、 干扰幅度和 计算时间的比较, 确定比较理想的出口边界条件.国家自然科学基金(编号K16017)资助项

OPEN BOUNDARY CONDITIONS IN SIMULATION BYSPECTRAL ELEMENT METHODS OF POISEUILLE-BENARD CHANNEL FLOW

研究二维矩形管道中底部加热的不可压缩Ｐｏｉｓｅｕｉｌｌｅ－Ｂｅｎａｒｄ流的谱元法数值计算问题．讨论各种不同的出口边界条件的处理及其对谱元法数值模拟的影响．通过干扰区、干扰幅度和计算时间的比较，确定比较理想的出口边界条件．2D simulation of Poiseuille-Benard channel flow by a spectral element method isperformed. The main purpose is to compare the effect of different open boundary conditions(OBCs) upon simulation results. A new boundary condition is applied in the context of spectralresolution, for which a new treatment technique is used. The computation are carried out forRe=10, Ri=150 and Pr=2/3. Among selected OBCs, a so-called Orlanski-type OBCs isproven to have better behavior as compared with the other OBCs.The choices of OBCs depend on the numerical methods to be used in the computation. Givena boundary condition, different treatment is required by different numerical methods. Four typesof OBCs are considered in this paper: (1) periodic condition; (2) Dirichlet condition, =0; (3)Neumann condition, =0; (4) Orlanski-type condition, = 0, where arethe related variables (velocity or temperature in this paper), V is a constant to determine, n isthe outward normal. Orlajnski-type OBCs is a non standard boundary condition in the contextof spectral approximation. It can be viewed as an approach, based on the following hypothesis:diffusion or pressure gradient or the sum of both is small at the outlet, the velocity is thereforeapproximately equal to the one situated on the characteristics. In other words, Orlanski-type OBCsis a generalized Dirichlet boundary condition which is adjusted following the evolution in time ofthe flow. This motivates us to apply a characteristics method to treating the Orlanski OBCs.Classical characteristics methods, implemented in the spectral approximation, cause however somedifficulties: it depends on the exact localization of the characteristic foots and on the polynomialinterpolation. These two procedures are generally expensive and possibly instable in the case ofhigh order numerical methods. These considerations lead us to introduce a local interpolationtechnique, proved numerically stable and of high precision.The Uzawa algorithm has been used to solve the resulting discrete equations stemming fromthe spectral method. The Uzawa ajgorithm is more efficient in terms of computational complexityand memory requirement than a direct approach, but it is sensitive on the properties of the algebraicsystems. We not only analyze OBCs effects to the computational results, but also compare theCPU time needed by different. QBCs. Numerical.results show that Orlanski-type OBCs gives moreaccurate results than the other OBCs. Thanks to the local characteristics method, Orlanski-typeOBCs maintains the symmetry property of the decoupled pressure system, and hence well adaptedto the conjugate gradient iterative procedtire, which is not the case for the Neumann-type OBCs.国家自然科学基金!K1601

不同剂量C离子注入Si单晶中Si_(1-x)C_x合金的形成及其特征

室温下在单晶Si中注入(0.6-1.5)%的C原子，利用高温退火固相外延了Si_(1-x)C_x合金，研究了不同注入剂量下Si_(1-x)C_x合金的形成及其特征，如果注入C原子的浓度小于0.6%，在850-950℃退火过程中，C原子容易与注入产生的损伤缺陷结合，难于形成Si_(1-x)C_x合金相。随注入C原子含量的增加，C原子几乎全部进入晶格位置形成Si_(1-x)C_x合金，但如果注入C原子的浓度达到1.5%，只有部分C原子参与形成Si_(1-x)C_x合金。升高退火温度，Si_(1-x)C_x合金相基本消失

不同剂量C离子注入Si单晶中Si_(1-x)C_x合金的形成及其特征

室温下在单晶Si中注入 (0 6— 1 5 ) %的C原子 ,利用高温退火固相外延了Si1-xCx 合金 ,研究了不同注入剂量下Si1-xCx 合金的形成及其特征 .如果注入C原子的浓度小于 0 6 % ,在 85 0— 95 0℃退火过程中 ,C原子容易与注入产生的损伤缺陷结合 ,难于形成Si1-xCx 合金相 .随注入C原子含量的增加 ,C原子几乎全部进入晶格位置形成Si1-xCx 合金 ,但如果注入C原子的浓度达到 1 5 % ,只有部分C原子参与形成Si1-xCx 合金 .升高退火温度 ,Si1-xCx 合金相基本消失