Fredholm-Volterra integral equation with singular kernel

The purpose of this paper is to obtain the solution of Fredholm-Volterra integral equation with singular kernel in the space L₂(?1, 1) × C(0,T), 0 ≤t ≤T < ∞, under certain conditions. The numerical method is used to solve the Fredholm integral equation of the second kind with weak singular kernel using the Toeplitz matrices. Also, the error estimate is computed and some numerical examples are computed using the MathCad package.     

一类具一阶奇性核的奇异积分方程

当(?)是复平面C上的光滑封闭曲线,k（z）是在(?)所围成的有界闭区域上连续.在其内部解析的函数时.借助于奇异积分算子的广义逆.讨论了具一阶奇性核的正则型奇异积分方程:　在H类中的求解问题.作为应用,作者给出了当k（z）是一类有理函数时的具体解法,从而统一并推广了 Cauchy核和Hilbert核奇异积分方程的经典结果.     

Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel

This paper is concerned with a trigonometric Hermite wavelet Galerkin method for the Fredholm integral equations with weakly singular kernel. The kernel function of this integral equation considered here includes two parts, a weakly singular kernel part and a smooth kernel part. The approximation estimates for the weakly singular kernel function and the smooth part based on the trigonometric Hermite wavelet constructed by E. Quak [Trigonometric wavelets for Hermite interpolation, Math. Comp. 65 (1996) 683–722] are developed. The use of trigonometric Hermite interpolant wavelets for the discretization leads to a circulant block diagonal symmetrical system matrix. It is shown that we only need to compute and store O(N)$\mathrm{O}(N)$ entries for the weakly singular kernel representation matrix with dimensions N²$N^2$ which can reduce the whole computational cost and storage expense. The computational schemes of the resulting matrix elements are provided for the weakly singular kernel function. Furthermore, the convergence analysis is developed for the trigonometric wavelet method in this paper.     

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

In this discussion, a new numerical algorithm focused on the Haar wavelet is used to solve linear and nonlinear inverse problems with unknown heat source. The heat source is dependent on time and space variables. These types of inverse problems are ill-posed and are challenging to solve accurately. The linearization technique converted the nonlinear problem into simple nonhomogeneous partial differential equation. In this Haar wavelet collocation method (HWCM), the time part is discretized by using finite difference approximation, and space variables are handled by Haar series approximation. The main contribution of the proposed method is transforming this ill-posed problem into well-conditioned algebraic equation with the help of Haar functions, and hence, there is no need to implement any sort of regularization technique. The results of numerical method are efficient and stable for this ill-posed problems containing different noisy levels. We have utilized the proposed method on several numerical examples and have valuable efficiency and accuracy.     

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.     

Approximate solution of one singular integro-differential equation

In this paper we construct and theoretically justify a computational scheme for solving the Cauchy problem for a singular integro-differential equation of the first-order, where the integral over a segment of the real axis is understood in the sense of the Cauchy principal value. In addition, we study and solve approximately the integral equation with a special logarithmic kernel. We obtain uniform estimates for errors of approximate formulas. Orders of errors of approximate solutions are proved to be proportional to the order of the approximation error for the derivative of the density of the singular integral in the integro-differential equation.     

On the Application of a Generalization of Toeplitz Matrices to the Numerical Solution of Integral Equations with Weakly Singular Convolution Kernels

This paper discusses the numerical solution by product integrationof weakly singular Fredholm integral equations of the secondkind with symmetric difference kernels. The product integrationmethod uses a piecewise polynomial, in general, at most, continuousat the knots. A main result of the paper is to show that, owingto the difference kernel, a highly patterned linear system ofequations arises if the knots are equally spaced. Specificallythe order-N coefficient matrix is block-Toeplitz or a generalization,and centrosymmetric. An algorithm to solve this linear systemin O(N²) operations is presented. Unfortunately the asymptotic rate of convergence of the productintegration solution is limited if a uniform mesh is used. Amethod to improve the rate of convergence, while retaining apatterned coefficient matrix, is described. This method involvessubtraction of the singularities in the solution induced bythe weakly singular kernel. A higher rate of convergence forthe modified product integration method is shown. These resultsare illustrated by application to an integral equation arisingin outdoor sound propagation.     

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…     

On the exceptional case of the characteristic singular equation with Cauchy kernel

We study the exceptional case of the characteristic singular integral equation with Cauchy kernel in which its coefficients admit zeros or singularities of complex orders at finitely many points of the contour. By reduction to a linear conjugation problem, we obtain an explicit solution formula and solvability conditions for this equation in weighted Hölder classes.     

An asymptotic method of solving transient dynamic contact problems

Using a special approximation in the complex plane of the symbol of the kernel of the contact-problem integral equation, an asymptotic form of its solution is constructed which is the fundamental solution of the transient dynamic plane contact problem of the impact of a rigid punch with an elastic half-plane for short interaction times. The proposed approximation of the kernel symbol enables it to be approximated in the complex plane with any previously specified accuracy. Unlike existing approaches [1, 2, etc.], the approximation of the kernel symbol of the integral equation employed here enables the solution of this problem to be obtained in the form of simple formulae not containing singular quadratures.     

Numerical evaluation for Cauchy type singular integrals on the interval

The singular integral (SI) with the Cauchy kernel is considered. New quadrature formulas (QFs) based on the modification of discrete vortex method to approximate SI are constructed. Convergence of QFs and error bounds are shown in the classes of functions H^α([−1,1]) and C¹([−1,1]). Numerical examples are shown to validate the QFs constructed.     

Degenerate kernel schemes by wavelets for nonlinear integral equations on the real line

A degenerate kernel scheme is developed for nonlinear integral equations on the real line by approximation of kernels by wavelets. The rate of convergence of the approximate solutions is established in terms of the decay rate of the kernels andthe numbers of dilations and shifts used in approximation of the kernels. For linear equations, the Haar wavelet approximation is used and a numerical example is included.     

The numerical solution of volterra intergral equations with singular kernels

Convergent starting procedures for nonlinear Volterra integral equations with singularities in the kernel are considered. These methods may be used to solve the problem on the entire interval of integration and for proper choice of the nodes areA-stable. Some numerical examples are presented.     

A main theorem of spectral relationships for Volterra–Fredholm integral equation of the first kind and its applications

This paper is concerned to derive the main theorem of spectral relationships of Volterra–Fredholm integral equation (V‐FIE) of the first kind in the space L₂[?1,1]×C[0,T], ?1?x?1, 0?t?T<1. The Fredholm integral (FI) term is considered in position and its kernel takes a logarithmic form multiplying by a continuous function. While Volterra integral (VI) term in time with a positive continuous kernel. Many important special and new cases can be established from the main theorem. Moreover, we use it to solve V‐FIE of the second kind in the same space. The numerical results are computed and the error is calculated using Maple 12. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical methods for Fredholm integral equations with singular right-hand sides

Fredholm integral equations with the right-hand side having singularities at the endpoints are considered. The singularities are moved into the kernel that is subsequently regularized by a suitable one-to-one map. The Nyström method is applied to the regularized equation. The convergence, stability and well conditioning of the method is proved in spaces of weighted continuous functions. The special case of the weakly singular and symmetric kernel is also investigated. Several numerical tests are included.     

A discrete collocation method for Symm's integral equation on curves with corners

This paper is devoted to the approximate solution of the classical first-kind boundary integral equation with logarithmic kernel (Symm's equation) on a closed polygonal boundary in ℝ². We propose a fully discrete method with a trial space of trigonometric polynomials, combined with a trapezoidal rule approximation of the integrals. Before discretization the equation is transformed using a nonlinear (mesh grading) parametrization of the boundary curve which has the effect of smoothing out the singularities at the corners and yields fast convergence of the approximate solutions. The convergence results are illustrated with some numerical examples.     

Numerical method for the solution of integral equations in a problem with directional derivative for the Laplace equation outside open curves

By using a simple layer potential and an angular potential, one can reduce the problem with a directional derivative for the Laplace equation outside several open curves on the plane to a uniquely solvable system of integral equations that consists of an integral equation of the second kind and additional integral conditions. The kernel in the integral equation of the second kind contains singularities and can be represented as a Cauchy singular integral. We suggest a numerical method for solving a system of integral equations. Quadrature formulas for the logarithmic and angular potentials are represented. The quadrature formula for the logarithmic potential preserves the property of its continuity across the boundary (open curves).     

Convergence in the mean of the Fourier series in orthogonal polynomials

For weights p(t) and q(t) with a finite number of power-law-type singularities we obtain necessary and sufficient conditions for the inequality $$\parallel s_n^{(p)} (f)q\parallel _{L^\eta - (1,1)} \leqslant C\parallel fq\parallel _{L^\eta ( - 1,1)'}$$ to hold, where s_n ^(p)(f) is a partial sum of the Fourier series of the function f in terms of polynomials orthogonal on [?1, 1] with weight p(t). This inequality is used to solve the problem concerning convergence in the mean and also convergence almost everywhere of the partial sums s_n ^(p)(f).     

What is the complexity of periodic weakly singular integral equations?

A fast/quasifast solver for weakly singular periodic Fredholm integral equations is constructed in the situation where the information about the kernel and the free term of the equation is restricted to a finite number of sample values. Hence the complexity of weakly singular integral equations is the same as that for equations without singularities or close to that. AMS subject classification (2000)  65Y20, 65T05, 45B05     

A compact finite difference scheme for the fourth-order time-fractional integro-differential equation with a weakly singular kernel

In this paper, a compact finite difference scheme is constructed and investigated for the fourth-order time-fractional integro-differential equation with a weakly singular kernel. In the temporal direction, the Caputo derivative term is treated by means of L₁ discrete formula and the Riemann–Liouville fractional integral term is discretized by the second-order convolution quadrature rule. A fully discrete compact difference scheme is constructed with the space discretization by the fourth-order compact approximation. The stability and convergence are obtained by the discrete energy method, the Cholesky decomposition and the reduced-order method. Numerical experiments are presented to verify the theoretical analysis.