最优投影策略下解病态积分方程的快速迭代算法

基于最优的投影方法,构造了求解病态积分方程的截断快速Tikhonov迭代算法,与传统投影方法相比得到了相同的最优收敛率,但内积的计算个数少于传统投影方法.同时,给出了后验参数选择办法.算例证实了算法的有效性.     

L~p空间中第二类Fredholm积分方程一种投影数值解法

在L~p(1p∞)空间中对第二类Fredholm积分方程提出了一种新的投影算法,对积分算子进行均值投影,给出了算法的先验估计和后验估计.数值算例进一步验证了算法的合理性和有效性.     

积分方程方法在层状散射问题中的应用研究

本文研究声波在分层均匀介质中碰到不可穿透障碍物产生的混合边值散射问题. 应用边界积分方程法将原问题转化为与之等价的边界积分方程组, 通过分析积分算子的Fredholm性质, 得到正问题解的适定性. 应用Nystr\"om方法将积分算子离散, 给出远场模式的计算方法, 并利用具体的数值实验验证方法的有效性, 为进一步展开反问题的研究奠定理论基础.     

E2类二阶椭圆组一般形式的非线性边值问题

研究了E2类二阶椭圆型方程组相当广泛的一类非线性边值问题，通过引进一种代换把它化为一类非线性广义Riemann-Hilbert边值问题，再引进奇异积分算子，建立与该问题等价的非线性奇异积分方程。应用奇异积分算子性质和泛函分析与函数论方法，在一定的假设条件下，证得了该问题的可解性。     

一类奇异积分算子在加权空间的界的估计

设T是由Grubb和Moore引入的一类奇异积分算子,它的核满足一种新型利普希茨正则性.T^*是由T确定的极大奇异积分算子.本文通过建立与T和T^*相关的grand极大算子的弱型端点估计,得到了算子T和T^*在加权空间的由A_p权常数表示的界的估计和弱型端点估计.     

截断策略下求解原始数据均有扰动的第一类Fredholm积分方程的多尺度Galerkin方法

本文研究求解原始数据均有扰动的第一类Fredholm积分方程的多尺度截断快速算法,给出了改进的后验参数选择准则,证明了收敛率能达到最优,给出的例子说明了方法的有效性.     

沿复合曲线的粗糙核参数型Marcinkiewicz积分算子的有界性

研究沿复合曲线的粗糙核参数型Marcinkiewicz积分算子,在积分核满足相当弱的尺寸条件下,建立了这些算子的L~p有界性.作为应用,相应于面积积分和Littlewood-Paley-g_λ~*函数的参数型Marcinkiewicz积分算子的L~p有界性也被给出.     

闭子模的酉等价与遗传C*-子代数的稳定同构问题

在Hilbert C~*-模框架下,给出了闭子模之间的酉等价与相应的遗传C~*-子代数的*同构,及对应的开投影的等价性的关系定理.     

关于Hilbert积分算子的一个分解

应用权函数及参量化的方法,建立了一对新的Hilbert型积分不等式,它是Hilbert积分不等式的一个分解.还考虑了其等价式,引入参数的最佳推广式及逆式.作为应用,求出了Hilbert积分算子的一个范数分解式.     

拟齐次核的 Hilbert 型重积分不等式最佳搭配参数的等价条件及应用*

利用权函数方法, 讨论拟齐次核Hilbert型重积分不等式的最佳搭配参数,得到最佳搭配参数的若干等价条件及不等式最佳常数因子的表达公式. 最后讨论其在奇异积分算子理论中的应用.     

函数域的双重根式扩域及素除子的密度

设K/F_q是整体函数域,l是与q互素的素数,ξ_1是K的固定代数闭包中的本原l次单位根.对于a,b∈K~*-(K~*)~l,本文主要讨论了根式扩域K(a~(1/2))与K(a~(1/l),(b~(1/l))的性质,利用Kummer理论给出了K(a~(1/l))/K与K(a~(1/l),b~(1/l))/K不是几何扩张的充要条件.当a,b是l-无关时,对于K的素除子P及对应的离散赋值环θ_P,利用这两类扩张的性质,通过分析a,b生成循环群(θ_P/P)~*的充要条件,本文明确给出了满足使得a,b生成循环群(θ_P/P)~*的全体素除子集合M_(a,b)的Dirichlet密度公式.     

求解Lavrentiev迭代方程的多尺度快速配置算法

考虑了第一类Fredholm积分方程的求解.采用有矩阵压缩策略的多尺度配置方法来离散Lavrentiev迭代方程,在积分算子是弱扇形紧算子时,给出近似解的先验误差估计,并给出了改进的后验参数的选择方法,得到了近似解的收敛率.最后,举例说明算法的有效性.     

The general equations of analytical dynamics

It is shown that the generalized Poincaré and Chetayev equations, which represent the equations of motion of mechanical systems using a certain closed system of infinitesimal linear operators, are related to the fundamental equations of analytical dynamics. Equations are derived in quasi-coordinates for the case of redundant variables; it is shown that when an energy integral exists the operator X₀ = ∂/∂t satisfies the Chetayev cyclic-displacement conditions. Using the energy integral the order of the system of equations of motion is reduced, and generalized Jacobi-Whittaker equations are derived from the Chetayev equations. It is shown that the Poincaré-Chetayev equations are equivalent to a number of equations of motion of non-holonomic systems, in particular, the Maggi, Volterra, Kane, and so on, equations. On the basis of these, and also of other previously obtained results, the Poincaré and Chetayev equations in redundant variables, applicable both to holonomic and non-holonomic systems, can be regarded as general equations of classical dynamics, equivalent to the well-known fundamental forms of the equations of motion, a number of which follow as special cases from the Poincaré and Chetayev equations.     

On systems of linear functional equations of the second kind in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

We consider a general system of functional equations of the second kind in L ₂ with a continuous linear operator T satisfying the condition that zero lies in the limit spectrum of the adjoint operator T*. We show that this condition holds for the operators of a wide class containing, in particular, all integral operators. The system under study is reduced by means of a unitary transformation to an equivalent system of linear integral equations of the second kind in L ₂ with Carleman matrix kernel of a special kind. By a linear continuous invertible change, this system is reduced to an equivalent integral equation of the second kind in L ₂ with quasidegenerate Carleman kernel. It is possible to apply various approximate methods of solution for such an equation.     

On the approximate solution of some integral equations of the theory of elasticity and mathematical physics PMM vol. 31, no. 6, 1967, pp. 1117–1131

In this paper the development of the method presented in [1] is carried out with application to two types of integral equations encountered in mathematical physics in the investigation of many mixed problems with circular separation line of boundary conditions and in the investigation of plane mixed problems.

The algorithm is given for reducing these integral equations to solution of equivalent infinite linear algebraic systems. It is proved that the resulting infinite systems are quasi completely regular for sufficiently large values of dimensionless parameter A which enters into the systems. It is shown that reduction (truncation) of infinite systems results in finite systems of linear algebraic equations with almost triangular matrices. The last circumstance simplifies considerably the solution of these finite systems after which the solution of initial integral equations is found from simple equations. For given accuracy of the approximate solution and decrease of parameter λ the number of equations in reduced systems increases.

As an example the solution is presented for the axisymmetric problem of a die acting on an elastic layer lying without friction on a rigid foundation.     

各向异性EQ1rot非协调元高精度分析的一般格式

本文在各向异性网格下讨论了一般二阶椭圆方程的EQ₁^rot非协调有限元逼近. 利用Taylor展开, 积分恒等式和平均值技巧导出了一些关于该元新的高精度估计. 再结合该元所具有的二个特殊性质: (a)当精确解属于H³时, 其相容误差为O(h²)阶比它的插值误差高一阶; (b)插值算子与Ritz投影算子等价,得到了在能量模意义下O(h²)阶的超逼近性质. 进而,借助于插值后处理技术给出了整体超收敛的一般估计式.     

A composite collocation method for the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations

This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations. The method is based on the composite collocation method. The properties of hybrid of block-pulse functions and Lagrange polynomials are discussed and utilized to define the composite interpolation operator. The estimates for the errors are given. The composite interpolation operator together with the Gaussian integration formula are then used to transform the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations into a system of nonlinear equations. The efficiency and accuracy of the proposed method is illustrated by four numerical examples.     

一类非线性算子方程的正解问题

研究算子方程X^s+A^*X^-tA=Q的正算子解的存在性问题,通过构造有效的迭代序列,给出了算子方程X^s+A^*X^-tA=Q有正算子解的一些充分条件和必要条件,同时给出了该方程有极大解和唯一解的条件.     

变量核奇异积分和分数次微分加权范不等式

用T和D^γ（0 ≤ γ ≤ 1）分别表示变量核奇异积分和分数次微分算子.T^*和T^#分别为T的共轭算子及拟共轭算子.利用球调和多项式展式，本文得到了TD^γ-D^γT和（T^*-T^#）D^γ在?^q，λ_ω（Rⁿ）上的有界性.同时也得到了变量核奇异积分的积T₁T₂和拟积T₁°T₂的加权范不等式.