一类具有Riemann-Liouville分数阶积分条件的分数阶微分方程边值问题

研究了一类具有Riemann-Liouville分数阶积分条件的新分数阶微分方程边值问题,其非线性项包含Caputo型分数阶导数.将该问题转化为等价的积分方程,应用Leray-Schauder不动点定理结合一个范数形式的新不等式,获得了解的存在性充分条件,推广和改进了已有的结果,并给出了应用实例.     

一类具有Riemann-Liouville分数阶积分边值条件的奇异分数阶微分方程解的存在性

研究一类具有Riemann-Liouville分数阶积分边值条件的奇异分数阶微分方程解的存在性,其非线性项包含Caputo型分数阶导数,且在t=0具有奇异性.应用Schauder不动点定理获得了解的存在性定理,并给出了应用实例.     

一类特殊的Riemann-Liouville分数阶积分的几何意义

主要研究阶数等于0.5的Riemann-Liouville (RL)分数阶积分的几何意义.通过变量替换,证明了0.5阶RL积分等价于沿着圆弧的曲线积分.对于同一个积分,如果沿着曲线观察,此积分是曲线积分;如果沿着坐标轴观察,此积分是分数阶积分.此结论说明,分数阶微积分提供了一种观察客观世界的新角度.另外,应用分数阶理论求解了一个关于转动惯量的问题,此实例说明分数阶现象是广泛存在的.     

具有右侧Riemann-Liouville分数阶导数的分数阶问题极值解的存在性

研究了一类具有右侧Riemann-Liouville分数阶导数的分数阶问题的可解性.利用Banach不动点定理和单调迭代技术,得到了一些解存在的结果.最后,通过实例说明了主要结果.     

一类含高阶Riemann-Liouville型分数阶导数脉冲微分方程的通解

本文首先运用迭代法获得一类含多项Riemann-Liouville型分数阶导数的微分方程的连续通解,然后应用数学归纳法得到这类脉冲微分方程的分片连续通解. 所得结果归结于脉冲分数阶微分方程领域,对分数阶微分方程研究者有参考意义.     

含三个Riemann-Liouville分数阶导数的脉冲Langevin型方程的边值问题的可解性

设n,l,k为正整数且α∈(n-1,n),β∈(l-1,l),γ∈(k-1,k).该文首先利用迭代方法给出具有三个分数阶导数的Langevin方程[D₀^α+D₀₊^β-λD₀₊^γ]x(t)=P(t)的连续通解.然后,该文使用数学归纳法获得脉冲分数阶Langevin方程[D₀^α+D₀₊^β-λD₀₊^γ]x(t)=P(t),t∈(t_i,t_i+1],i∈N₀^m分片连续通解.接下来,该文运用获得的结果研究具有三个分数阶导数α,β∈(1,2),γ∈(0,1)的非线性脉冲Langevin方程的一类边值问题,通过将其化为积分方程,运用不动点定理建立这类边值问题解的存在性定理.最后,该文给出例子说明了主要结果的应用.     

一类具有分数阶积分条件的分数阶微分方程组边值问题的可解性

研究一类具有Riemann-Liouville分数阶积分条件的分数阶微分方程组边值问题,将该问题转化为等价的积分方程组,应用Leray-Schauder不动点定理和Banach压缩映像原理,结合一个分数阶形式的新不等式,获得了该问题解的存在性和唯一性结果,并给出一个应用实例.     

一类分数阶微分方程组积分边值问题的正解

利用锥拉伸和压缩不动点定理,研究了一类具有Riemann-Liouvile分数阶积分条件的分数阶微分方程组边值问题.结合该问题相应Green函数的性质,获得了其正解的存在性条件,并给出了一些应用实例.     

时间分数阶偏微分方程的数值算法研究

提出了一种求解时间分数阶偏微分方程的数值算法.根据方程的初值条件和边界条件构造辅助函数,应用辅助函数将方程转化为具有零初值条件和零边界条件的方程.应用差分算法计算方程中的分数阶导数,然后求解出方程的数值解.方程的分数阶可以为任意正实数,无论分数阶取何值,此算法均采用同样的差分格式求解方程.给出的两个计算实例说明此算法是...     

带有Riemann-Liouville型分数导数的分数阶边值问题的正解

本文讨论了带有Riemann-Liouville型分数导数的分数阶边值问题正解的存在性,针对非线性项满足一些不等式的条件下,运用不动点指数定理(缺方向性和同伦不变性)获得了两个存在性定理.     

Algorithms for Computing Characters for Symmetric Spaces

Symmetric spaces or more general symmetric k-varieties can be defined as the homogeneous spaces G _k /K _k , where G is a reductive algebraic group defined over a field k of characteristic not 2, K the fixed point group of an involution θ of G and G _k resp. K _k the sets k-rational points of G resp. K. These symmetric spaces have a fine structure of root systems, characters, Weyl groups etc., similar to the underlying algebraic group G. The relationship between the fine structure of the symmetric space and the group plays an important role in the study of these symmetric spaces and their applications. To develop a computer algebra package for symmetric spaces one needs explicit formulas expressing the fine structure of the symmetric space and group in terms of each other. In this paper we consider the case that k is algebraically closed and give explicit algorithmic formulas for expressing the characters of the weight lattice of the symmetric space in terms of the characters of the weight lattice of the group. These algorithms can easily be implemented in a computer algebra package. The root system of the symmetric space can be described as the image of the root system of the group under a projection π derived from an involution θ on . This implies that . Using these formulas for the characters of each of these lattices we show that in fact . A.G. Helminck is partially supported by N.S.F. Grant DMS-0532140.     

快速配置法中数值积分的误差控制策略

We propose two error control techniques for numerical integrations in fast multiscale collocation methods for solving Fredholm integral equations of the second kind with weakly singular kernels. Both techniques utilize quadratures for singular integrals using graded points. One has a polynomial order of accuracy if the integrand has a polynomial order of smoothness except at the singular point and the other has exponential order of accuracy if the integrand has an infinite order of smoothness except at the singular point. We estimate the order of convergence and computational complexity of the corresponding approximate solutions of the equation. We prove that the second technique preserves the order of convergence and computational complexity of the original collocation method. Numerical experiments are presented to illustrate the theoretical estimates.     

Output-Sensitive Algorithms for Computing Nearest-Neighbour Decision Boundaries

Given a set R of red points and a set B of blue points, the nearest-neighbour decision rule classifies a new point q as red (respectively, blue) if the closest point to q in R B comes from R (respectively, B). This rule implicitly partitions space into a red set and a blue set that are separated by a red-blue decision boundary. In this paper we develop output-sensitive algorithms for computing this decision boundary for point sets on the line and in ². Both algorithms run in time O(n log k), where k is the number of points that contribute to the decision boundary. This running time is the best possible when parameterizing with respect to n and k.     

一类含有分数阶导数的二自由度耦合系统

研究了一类含有小扰动具有分数阶导数的二自由度耦合振子的振动问题.首先对含有由Riemann Liouville定义的分数阶导数的振动方程组构造渐近解,利用多重尺度法,得到振动问题的可解性条件.然后在可解性条件下,得到分数阶指数、系数及小参数对振动的影响,并求得渐近解.最后研究了该解的稳定性,发现定常解都是稳定的     

Numerical Analysis of Time Fractional-Order Unsaturated Flow and Its Application北大核心CSCD

非饱和渗流过程的数值模拟对土质边坡稳定性分析、地下污染物迁移模拟等众多领域有着重要的意义。Richards方程由于其普遍适用性被广泛地应用,然而Richards方程所描述的渗流过程并未考虑在自然环境和实验中存在的反常扩散现象。针对这一问题,该文结合Caputo导数得到了具有更广泛渗流意义的时间分数阶Richards方程,采用有限差分法得到其离散格式并采用Picard法迭代求解,以及对分数阶参数和土水特征曲线进行了敏感性分析。最后,结合土柱入渗实验数据,比较了不同土水特征曲线下时间分数阶Richards方程得到的数值解。结果表明,VGM模型的时间分数阶Richards方程与实测数据具有更好的拟合效果,能够更好地描述地下水在非饱和土中的渗流过程。     

定积分计算方法及其数值试验

以定积分概念所蕴含的数学思想和哲学思想为指导,给出定积分近似计算的两种方法,即定义法和蒙特卡罗方法,并借助Matlab软件编程对其加以验证,数值试验结果显示所给方法可行且有效．     

Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps

We provide explicit and efficient reduction algorithms based on discrete Morse theory to simplify homology computation for a very general class of complexes. A set-valued map of top-dimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) continuous function, especially when the evaluation of that function is subject to measurement errors. We introduce a new Morse theoretic preprocessing framework for deriving chain maps from such set-valued maps, and hence provide an effective scheme for computing the morphism induced on homology by the approximated continuous function.     

Algorithms for Computing U(N) Clebsch Gordan Coefficients

If V ^(m) is an irreducible representation space for the unitary group U(N), then the r-fold tensor product space, is in general reducible. Such a reducible representation can be reduced to a direct sum of irreducible representation spaces, albeit with multiplicity. Clebsch–Gordan coefficients are the overlap coefficients between an orthonormal basis in the tensor product space and an orthonormal basis in the direct sum space. Since such coefficients are basis dependent, bases in the U(N) irrep spaces must be chosen. In this paper we use the Gelfand–Zetlin basis, built out of the chain of subgroups U(N)⊃...⊃U(1). Our first result is to derive algorithms for generating Gelfand–Zetlin bases from Gelfand–Zetlin tableaux. Given these concrete basis realizations we develop algorithms for computing the Clebsch–Gordan coefficients themselves.