Numerical Methods of the Fractional Kinetic Equations and Theoretical Analysis

分数阶动力方程近年来得到广泛的兴趣和关注。其主要原因是由于分数阶微积分理论自身的迅速发展，以及其在物理、化学、生物，环境科学，工程以及金融等各类学科中的广泛应用。分数阶动力方程为描述不同物质的记忆和继承性质提供了强有力的工具。然而，分数阶动力方程的解析解是比较复杂的，多数解析解都包含了有级数形式或特殊函数。而且，多数分数阶动力方程的解不能显式地得到。这就促使我们必须考虑有效的数值方法。目前，关于分数阶动力方程的数值方法以及相关的稳定性和收敛性分析相当有限，而且很难得到。这些激励我们发展有效的数值方法解分数阶的微分方法。在本论文中，我们考虑两种类型的分数阶动力方程。第一类分数阶动力方程是带有扩散...Fractional kinetic equations have been of great interest recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, chemistry, biology, environmental sciences, engineering and finance. Fractional kinetic equations provide a powerful instrument for the description of memory a...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：1702005140300

时间多项分数阶波动-反应方程的无网格方法

尝试采用基于无网格方法的移动最小二乘求解带有时间多项分数阶导数的波动-反应方程.首先利用差分思想离散多项时间分数阶导数,并用移动最小二乘法离散空间变量,得到微分方程的数值逼近..

LBO的腔内倍频特性研究

从倍频转换效率公式和腔内倍频的稳定态条件出发,得到倍频波的功率密度公式,进而得出倍频晶体最佳长度和倍频波最大功率密度的表达式。以LBO晶体Ⅰ类临界相位匹配腔内倍频946 nm为例,根据倍频波的功率密度公式,从理论上讨论了LBO长度、功率密度比对倍频波功率密度的影响,为LBO倍频产生473 nm蓝光实验提供了理论指导。虽然这里只是讨论了LBOⅠ类临界相位匹配倍频946 nm,但对所有腔内倍频实验(不同倍频晶体或不同倍频频率)具有借鉴作用

Relationship between mannose-binding protein polymorphism and patients with liver cirrhosis and hepatocellular carcinoma

目的探讨甘露糖结合蛋白(MbP)基因突变与肝硬化及肝癌的关系。方法采用聚合酶链式反应-限制性片段长度多态性(PCr-rflP)方法和实时荧光定量PCr(fQ-PCr)技术针对代偿性肝硬化(CC)患者73例、失代偿性肝硬化(dC)患者78例、肝细胞癌(HCC)患者35例和对照组88例健康者的MbP基因第54位密码子多态性进行检测。结果 HCC组的MbP基因ggC/gAC基因型频率和gAC等位基因频率与对照组比较,差异无统计学意义(P>0.05)。CC组、dC组MbP基因ggC/gAC基因型频率和gAC等位基因频率均显著高于HCC组和对照组(P 0.05).GAC allele frequency was also highest prevalence (36.5%) in DC group than that in CC group and HCC group (P < 0.05).Conclusions The MBP codon 54 polymorphism is associated with the progression of liver cirrhosis and might not play an important role in the development of hepatocellular carcinoma.福建省漳州市科技计划资助项目(Z2010085

U n ifo rm Second2O rder A ccu rate D ifference Schem es fo r NonSelf2adjo in t Singu lar Pertu rbat ion P rob lem in Con servat ion Form

摘要　考虑带小参数的守恒型非自共轭奇异摄动问题, 建立并证明一个二阶一致收格式. Abs tra c t　Non self2adjo in t singu lar pertu rbat ion p rob lem in con servat ion fo rm is discu ssed. A un ifo rm ly second2o rder accu rate difference schem e is p resen ted and p roven

Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process

In this paper, we consider the following non-linear fractional reaction–subdiffusion process (NFR-SubDP): Formula where f(u, x, t) is a linear function of u, the function g(u, x, t) satisfies the Lipschitz condition and 0Dt1–{gamma} is the Riemann–Liouville time fractional partial derivative of order 1 – {gamma}. We propose a new computationally efficient numerical technique to simulate the process. Firstly, the NFR-SubDP is decoupled, which is equivalent to solving a non-linear fractional reaction–subdiffusion equation (NFR-SubDE). Secondly, we propose an implicit numerical method to approximate the NFR-SubDE. Thirdly, the stability and convergence of the method are discussed using a new energy method. Finally, some numerical examples are presented to show the application of the present technique. This method and supporting theoretical results can also be applied to fractional integrodifferential equations

An Effective Numerical Method of the Rayleigh-Stokes Problem for a Heated Generalized Second Grade Fluid With Fractional Derivative

考虑加热下分数阶广义二阶流体的rAylEIgH-STOkES问题(rSP-HgSgf),提出了一种逼近有界区域内rSP-HgSgf的有效数值方法.并且讨论了所提出方法的稳定性和收敛性.最后,利用数值例子体现数值方法的有效性.The Rayleigh-Stokes problem for a heated generalized second grade fluid(RSP-HGSGF) with fractional derivative was considered.An effective numerical method for approximating RSP-HGSGF in a bounded domain was presented.And the stability and convergence of the numerical method were analyzed.Finally,some numerical examples were presented to show the application of the present technique

含双参数半线性奇异摄动问题的L~1一致收敛差分格式

A nonlinear difference scheme is given for solving a semilinear singular perturbation problem involving two parameters. The solution of the scheme is shown to be of first order accuracy in the discrete L1 norm, uniformly in the perturbation parameter. A numerical example is given.福建省自然科学墓金资助项

AN EXPLICIT APPROXIMATION FOR THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION

1 引言分数阶微分方程产生于一些反常扩散模型,已经被利用于模拟在工程,物理,化学和其它科学领域的许多现象．目前已有许多研究专家学者[1][2][3][4]从理论上对方程进行了研究．数值解方面,刘发旺教授等[5,6]首先提出利用行方法求解空间分数阶扩散方程来In this paper, a space-time fractional diffusion equation is considered. An explicit approximation is constructed. Stability and convergence of the method are discussed. Finally, some numerical examples are presented to show the space-time diffusion behaviors.国家自然科学基金(10271098) 福建省自然科学基金(Z0511009)赞助

L ∞U n ifo rm Convergence of a D ifference Schem e fo r a Singu lar Pertu rbat ion P rob lem Invo lving Two Param eters

摘要　对带有两个小参数的奇异摄动问题, 给出一种差分格式. 并证明其在L ∞ 范数意义下 的一致收敛性. 最后给出数值例子. 　Abs tra c t　A difference schem e is given fo r so lving a singu lar pertu rbat ion p rob lem invo lving two param eters. The so lu t ion of the schem e is show n to be of un ifo rm convergences in the discrete L ∞ no rm. The num erical examp les are given