一类对偶积分方程组正则化为Cauchy奇异积分方程组解法

本文基于Mellin变换法求解复杂更一般形式的对偶积分方程组．通过积分变换，由实数域化成复数域上的方程组，引入未知函数的积分变换，移动积分路径，应用Cauchy积分定理，实现退耦正则化为Cauchy奇异积分方程组，由此给出一般性解，并严格证明了对偶积分方程组退耦正则化为Cauchy奇异积分方程组与原对偶积分方程组等价性，以及对偶积分方程组解的存在性和唯一性．给出的解法和理论解，作为求解复杂对偶积分方程组一种有效解法，可供求解复杂的数学、物理、力学中的混合边值问题应用．     

一类多节对偶积分方程组正则化为超(强)奇异积分方程组求解法

基于Mellin变换法,首先方程组进行Mellin变换,然后,通过引入新的未知函数的Mellin变换代换原来未知函数的Mellin变换,使对偶积分方程组退耦正则化为超(强)奇异积分方程组．将未知函数分解并表示成未知函数和已知幂函数的乘积,幂指数(a_i,v_i)需使超(强)奇异积分方程组中的超(强)奇异积分,在端点(a_i,b_i)有界或可积奇异,求解超(强)奇异积分方程组可以使用有限部分积分式．将未知函数展成任意完备函数系(?)_n*(u)的级数,将超(强)奇异积分方程组,化成线性代数方程组,通过求解级数中的各项系数,由此给出对偶积分方程组的一般性解．并严格证明了对偶积分方程组和由它化成的超(强)奇异积分方程组的等价性,解的存在性和解的表示形式不唯一性．本文给出的理论解和解法,可供求解数学,物理,力学中的混合边值问题应用．     

一般形式的奇异对偶积分方程组正交多项式求解法

基于Jacobi正交多项式法,直接求解一般形式的对偶积分方程组,将对偶积分方程组中的未知函数,表示成n次Jacobi正交多项式级数,用正交多项式将奇异对偶积分方程组,化成线性代数方程组,通过求解级数中的各项系数,由此给出奇异对偶积分方程组的一般性解,并严格证明了奇异对偶积分方程组和由它化成的线性代数方程组的等价性,解的存在性和解的表示形式不唯一性．本文给出的理论解和解法,可供求解复杂的数学、物理、软科学中的混合边值问题应用．     

一类对偶积分方程组正则化为第一类含对数核的Fredholm奇异积分方程组解法

引入辅助未知函数及辅助未知函数的积分关系式,表示原未知函数,将对偶积分方程组退耦.应用Sonine第一有限积分公式,实现化为Abel型积分方程组,应用Abel反演变换并化简,正则化为含对数核的第一类Fredholm奇异积分方程组.由此给出奇异积分方程组的一般性解,进而获得对偶积分方程组的解析解,同时严格地证明了,对偶积分方程组和由它化成的含对数核的奇异积分方程组的等价性,以及对偶积分方程组解的存在性和唯一性.     

压电材料Ⅰ型裂纹动态问题的对偶方程组及其求解

首先引入势函数,用势函数表示压电材料的基本微分方程,并采用Laplace变换、半无限对称Fourier正弦变换和Fourier余弦变换,对微分方程进行变换和初步求解;然后通过Fourier反演和引入边界条件,建立了二维压电材料动态裂纹问题的对偶方程组; 再根据Bessel函数性质, 利用Abel型积分方程及其反演,将对偶方程组化为第二类Fredholm积分方程组．结果表明,方法是可行的,可以成为研究此类问题的一种有效方法．     

非线性方程组的一种解法

文中给出了求非线性方程组的一种解法；它找到一个变换，可使原方程组变换成一个新的方程组，通过求变换后方程组的解求得原方程组的解．     

第二类Fredholm积分方程组的分段泰勒级数展开法

积分方程出现在数学物理的各种问题中,寻求其简单而又有效的解法显得很有必要.提出一种求解第二类线性Fredholm积分方程组的新解法,利用分段泰勒级数展开,通过引入两个参数得到近似解的表达式,并对近似解的收敛性和误差进行分析.通过与已有数值方法的比较,说明此方法的可行性和有效性。     

一类非线性偏微分方程组的直接解法

根据Hopf-Cole变换法和试探函数法的基本思想,引入一个变换,并把它应用于求解(2+1)维破裂孤子方程组、(2+1)维Nizhnik-Novikov-Vesslov方程组和(2+1)维Broer-Kaup方程组,得到了这三个方程组的许多新的解析解,包括孤波解和奇异行波解.该方法也适用于其它方程组.     

一类奇异积分方程组的样条间接近似解法

本文利用三次复插值样条函数给了定义于复平面上光滑封闭曲线上的一类奇异积分方程组（１）的一种间接近似解法，讨论了误差估计和一致收敛性。     

对偶积分方程与对偶积分方程组及其在固体力学与流体力学中的某些应用

本文首先简要地介绍了文献[1、2]关于对偶积分方程的解,在某些实际问题中,出现的是更为复杂的时偶积分方程组。在文献[1、2]的启发下,我们把这种积分方程组化成复数域上的一般函数方程组,并且由此给出形式解。然后介绍我们用上述两种理论计算得到的固体力学与流体力学中某些混合边值问题的实例,其中出现的对偶积分方程组,用本文建议的方法,得到了精确解。     

Existence of solutions for dual singular integral equations with convolution kernels in case of non-normal type

This paper is devoted to the study of dual singular integral equations with convolution kernels in the case of non-normal type. Via using the Fourier transforms, we transform such equations into Riemann boundary value problems. To solve the equation, we establish the regularity theory of solvability. The general solutions and the solvable conditions of the equation are obtained. Especially, we investigate the asymptotic property of solutions at nodes. This paper will have a significant meaning for the study of improving and developing complex analysis, integral equations and Riemann boundary value problems.     

A COLLOCATION METHOD FOR THE CONDUCTIVITY PROBLEM WITH DISCONTINUOUS COEFFICIENT

In this paper, a new collocation BEM for the Robin boundary value problem of the conductivity equation ▽(γ▽u) = 0 is discussed, where the 7 is a piecewise constant function. By the integral representation formula of the solution of the conductivity equation on the boundary and interface, the boundary integral equations are obtained. We discuss the properties of these integral equations and propose a collocation method for solving these boundary integral equations. Both the theoretical analysis and the error analysis are presented and a numerical example is given.     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

An extrapolation method for a class of boundary integral equations

Boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using Green's formulas. For solving the induced boundary integral equations, a Nyström scheme and its extrapolation method are derived for periodic Fredholm integral equations of the second kind with logarithmic singularity. Asymptotic expansions for the approximate solutions obtained by the Nyström scheme are developed to analyze the extrapolation method. Some computational aspects of the methods are considered, and two numerical examples are given to illustrate the acceleration of convergence.

     

The solvability of some kinds of singular integral equations of convolution type with variable integral limits

In this paper, we discuss several classes of convolution type singular integral equations with variable integral limits in class $ H^*_1 $. By means of the theory of complex analysis, Fourier analysis and integral transforms, we can transform singular integral equations with variable integral limits into the Riemann boundary value problems with discontinuous coefficients. Under the solvability conditions, the existence and uniqueness of the general solutions can be obtained. Further, we analyze the asymptotic properties of the solutions at the nodes. Our work improves the Noether theory of singular integral equations and boundary value problems, and develops the knowledge architecture of complex analysis.     

Approximate solutions of linear Volterra integral equation systems with variable coefficients

In this paper, a new approximate method has been presented to solve the linear Volterra integral equation systems (VIEs). This method transforms the integral system into the matrix equation with the help of Taylor series. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Also, this method gives the analytic solution when the exact solutions are polynomials. So as to show this capability and robustness, some systems of VIEs are solved by the presented method in order to obtain their approximate solutions.     

关于带复平移的奇异积分方程

本文得出了带多个上下复平移的奇异积分方程的等价方程,并由此给出了其可解的充分条件,同时获得了奇异分方程的解的具体表达形式。