求解一类具有Hibert核的奇异积分方程的小波方法

1　引　　言近年来,用小波方法数值求解积分方程越来越引起人们的注意.文献[1]提出的算法可将一类积分算子所对应的矩阵稀疏化,为小波方法快速求解积分方程开辟了一条新的道路这方面的研究不仅可以深入发展小波理论和应用算法,深入发展小波方法的功效,而且对边界元方法有重要的指导意义.然而研究稳健快速的数值方法,一直是这方面研究的难点问题.本文考虑带Hilbert核的奇异积分方程q(y)=12π∫2π0f(x)ctg12(x-y)dx,y∈[0,2π],(1.1)的小波数值解法;其中f(x)∈H2π,q(y)∈H2π是以2π为周期的Holder类函数;q(y)已知,f(x)待求解;(1.1)式右…     

Numerical solution of boundary integral equations by means of attenuation factors

We consider first-kind boundary integral equations with logarithmickernel such as those arising from solving Dirichlet problemsfor the Laplace equation by means of single-layer potentials.The first-kind equations are transformed into equivalent equationsof the second kind which contain the conjugation operator andwhich are then solved with a degenerate-kernel method basedon Fourier analysis and attenuation factors. The approximationswe consider, among them spline interpolants, are linear andtranslation invariant. In view of the particularly small kernel,the linear systems resulting from the discretization can besolved directly by fixed-point iteration.     

A New defy for iteration methods

The work presents an adaptation of iteration method for solving a class of thirst order partial nonlinear differential equation with mixed derivatives.The class of partial differential equations present here is not solvable with neither the method of Green function, the most usual iteration methods for instance variational iteration method, homotopy perturbation method and Adomian decomposition method, nor integral transform for instance Laplace,Sumudu, Fourier and Mellin transform. We presented the stability and convergence of the used method for solving this class of nonlinear chaotic equations.Using the proposed method, we obtained exact solutions to this kind of equations.     

Mixed problem for pseudoparabolic integro-differential equation with degenerate kernel

A mixed problem for a certain nonlinear third-order intregro-differential equation of the pseudoparabolic type with a degenerate kernel is considered. The method of degenerate kernel is essentially used and developed and the Fourier method of variable separation is employed for this equation. A system of countable systems of algebraic equations is first obtained; after it is solved, a countable system of nonlinear integral equations is derived. The method of sequential approximations is used to prove the theorem on the unique solvability of the mixed problem.     

Application of the Fourier Transform to the Solution of Singular Integral Equations

A method for solving a certain system of singular integral equations with constant coefficients is proposed. It is based on a procedure for reducing singular equations to equations with continuous difference kernel; the solution of the latter is constructed by using the classical Fourier transform in the class of absolutely integrable functions. Explicit expressions for the solution of the singular integral equations under consideration are obtained.     

An automatic augmented galerkin method for singular integral equations with Hilbert kernel

In [1, 2], described a Chebyshev series method for the numerical solution of integral equations with three automatic algorithms for computing two regularization parameters,C _f andr. Here we describe a Fourier series expansion method for a class of singular integral equations with Hilbert kernel and constant coefficients with using a new automatic algorithm.     

Higher-order quadratures for circulant preconditioned Wiener-Hopf equations

In this paper, we consider solving matrix systems arising from the discretization of Wiener-Hopf equations by preconditioned conjugate gradient (PCG) methods. Circulant integral operators as preconditioners have been proposed and studied. However, the discretization of these preconditioned equations by employing higher-order quadratures leads to matrix systems that cannot be solved efficiently by using fast Fourier transforms (FFTs). The aim of this paper is to propose new preconditioners for Wiener-Hopf equations. The discretization of these preconditioned operator equations by higher-order quadratures leads to matrix systems that involve only Toeplitz, circulant and diagonal matrix-vector multiplications and hence can be computed efficiently by FFTs in each iteration. We show that with the proper choice of kernel functions of Wiener-Hopf equations, the resulting preconditioned operators will have clustered spectra and therefore the PCG method converges very fast. Numerical examples are given to illustrate the fast convergence of the method and the improvement of the accuracy of the computed solutions with using higher-order quadratures.Research supported by the Cooperative Research Centre for Advanced Computational Systems.Research supported in part by Lee Ka Shing scholarship.     

Superconvergence results of iterated projection methods for linear Volterra integral equations of second kind

In this paper, we develop the iteration techniques for Galerkin and collocation methods for linear Volterra integral equations of the second kind with a smooth kernel, using piecewise constant functions. We prove that the convergence rates for every step of iteration improve by order \({\mathcal {O}}(h^{2})\) for Galerkin method, whereas in collocation method, it is improved by \({\mathcal {O}}(h)\) in infinity norm. We also show that the system to be inverted remains same for every iteration as in the original projection methods. We illustrate our results by numerical examples.     

N-group neutron transport theory: A criticality problem in slab geometry

The steady-state equation for N-group neutron transport in slab geometry is written as an integral equation. A spectral analysis is made of the integral operator and related to the criticality problem. The method depends on a representation for the resolvent kernel for a subcritical slab and on analytic continuation in a complex parameter to characterize eigenvalues in terms of singularities of the resolvent. The analytic continuation is based on a bifurcation analysis of some nonlinear matrix integral equations whose solutions provide a matrix Wiener-Hopf factorization of the Fourier transform of the kernel of the transport operator.     

On global representation of lagrangian distributions and solutions of hyperbolic equations

In this paper we develop a new approach to the theory of Fourier integral operators. It allows us to represent the Schwartz kernel of a Fourier integral operator by one oscillatory integral with a complex phase function. We consider Fourier integral operators associated with canonical transformations, having in mind applications to hyperbolic equations. As a by-product we obtain yet another formula for the Maslov index. © 1994 John Wiley & Sons, Inc.     

Analytic solutions to problems of radiative transport in an absorbing medium

A method is proposed for solving the linear integral equations governing some problems of radiation transport. The Fourier transform technique provides the solution as a convolution integral of the kernel and a series expansion in orthogonal functions. Two standard problems were considered as an application and the results given in comparison with those obtained by different approaches.     

轴对称问题的积分方程的迭代解法

将集度分别为x(ξ)和y(ξ)的集中力和挤压中心沿物体外弹性空间z轴分布,并迭加应力为常数项的解,就能使轴对称应力问题归结为两个联立的一维Fredholm第一种积分方程,本文研究此类方程的迭代解法.给出与E.Rakotch收缩映射定理等价的引理和迭代收敛证明.     

Dual integral equations related to the kontorovich-lebedev transform : PMM vol. 38, n 6, 1974, pp. 1090–1097

We study the dual integral equations related to the Kontorovich-Lebedev integral transforms arising in the course of solution of the problems of mathematical physics, in particular of the mixed boundary value problems for the wedge-shaped regions. We show that the solutions of these equations can be expressed in quadratures, using the auxilliary functions satisfying the integral Fredholm equation of second kind with a symmetric kernel.At present, the dual equations investigated in most detail are those connected with the Fourier and Hankel integral transforms. The results obtained and their applications are given in [1–3]. A large number of papers also deal with the theory and applications of the dual integral equations connected with the Mehler-Fock integral transform and its generalizations [4–11]., The dual integral transforms considered in the present paper belong to a more complex class than those listed above, and so far, no effective solution has been obtained for them. The only relevant results known to the authors are those in [12, 13]. In [12] a method of solving the equations (1.2) is given for a single particular value of the parameter γ = π/2, while in [13] the dual equations of the type under consideration are reduced to a solution of an infinite system of linear algebraic equations.     

The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations

Summary Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.     

Analytic solutions to problems of radiative transport in an absorbing medium

A method is proposed for solving the linear integral equations governing some problems of radiation transport. The Fourier transform technique provides the solution as a convolution integral of the kernel and a series expansion in orthogonal functions. Two standard problems were considered as an application and the results given in comparison with those obtained by different approaches.

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Riassunto Si propone un metodo per risolvere le equazioni lineari integrali del trasporto radiativo. La tecnica della trasformata di Fourier fornisce la soluzione come integrale di convoluzione del kernel e di uno sviluppo in serie di funzioni ortogonali. Come applicazione si sono risolti due problemi classici ed i risultati si sono confrontati con quelli ottenuti mediante altri procedimenti.

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This work was partially supported by the Consiglio Nazionale delle Ricerche, Gruppo di Fisica Matematica.     

一维热传导方程移动边界识别的计算方法

构造了一种正则化的积分方程方法来由Cauchy数据确定一维热传导方程的移动边界．在将区域延拓至规则区域后,通过Fourier方法将问题转化为一个第一类Volterra积分方程．然后分别用Lavrentiev正则化方法以及Tikhonov正则化方法将不稳定的第一类Volterra积分方程转化为适定的第二类积分方程,并分别将积分方程转化为常微分方程组,并用Runge—Kutta方法数值求解,以及直接离散来求解．最后通过自由边界上的条件得到数值的移动边界．通过一些数值试验表明此方法是有效可行的,并且给出的方法无需迭代,数值计算较简单．     

On a method for solving the inverse scattering problem on the line

A method for solving the inverse scattering problem on the line is proposed. It is based on a Fourier‐Laguerre series representation of the integral transmutation kernel. Substitution of the representation into the Gel'fand‐Levitan‐Marchenko equation leads to a linear algebraic system of equations and consequently to a simple algorithm for recovering the potential.     

积分算子本征值问题伽略金法的后验误差指示子

1引言 设G是R~n中有界域，积分算予Tx(s)=integral from n k(s.t)x(t)dt.(s∈G)是映L~2(G)到L~2(G)中的自共轭全连续算子。△={△}是G的拟一致部分。     

The transmutation operator method for efficient solution of the inverse Sturm‐Liouville problem on a half‐line

The inverse Sturm‐Liouville problem on a half‐line is considered. With the aid of a Fourier‐Legendre series representation of the transmutation integral kernel and the Gel'fand‐Levitan equation, the numerical solution of the problem is reduced to a system of linear algebraic equations. The potential q is recovered from the first coefficient of the Fourier‐Legendre series. The resulting numerical method is direct and simple. The results of the numerical experiments are presented.     

C编号Walsh函数及Manz算法

沃尔什编号Walsh函数是Walsh在1923年给出的;1931年,Paley定义了佩利编号Walsh函数;哈德玛编号Walsh函数是根据Hadamard 1893年的工作,由专用