The numerical solution of Fredholm integral equations of the second kind with singular kernels

A numerical method is given for integral equations with singular kernels. The method modifies the ideas of product integration contained in [3], and it is analyzed using the general schema of [1]. The emphasis is on equations which were not amenable to the method in [3]; in addition, the method tries to keep computer running time to a minimum, while maintaining an adequate order of convergence. The method is illustrated extensively with an integral equation reformulation of boundary value problems for u–P(r ²)u=0; see [9].This research was supported in part by NSF grant GP-8554.     

Product integration for volterra integral equations of the second kind with weakly singular kernels

We introduce a new numerical approach for solving Volterra integral equations of the second kind when the kernel contains a mild singularity. We give a convergence result. We also present numerical examples which show the performance and efficiency of our method.

     

A Jacobi-collocation method for solving second kind Fredholm integral equations with weakly singular kernels

In this work, we propose a Jacobi-collocation method to solve the second kind linear Fredholm integral equations with weakly singular kernels. Particularly, we consider the case when the underlying solutions are sufficiently smooth. In this case, the proposed method leads to a fully discrete linear system. We show that the fully discrete integral operator is stable in both infinite and weighted square norms. Furthermore, we establish that the approximate solution arrives at an optimal convergence order under the two norms. Finally, we give some numerical examples, which confirm the theoretical prediction of the exponential rate of convergence.     

A fast Fourier-Galerkin method for solving integral equations of second kind with weakly singular kernels

We propose in this paper a convenient way to compress the dense matrix representation of a compact integral operator with a weakly singular kernel under the Fourier basis. This compression leads to a sparse matrix with only ${\mathcal{O}}(n\log n)$ number of nonzero entries, where 2n+1 denotes the order of the matrix. Based on this compression strategy, we develop a fast Fourier-Galerkin method for solving second kind integral equations with weakly singular kernels. We prove that the approximate order of the truncated equation remains optimal and that the spectral condition number of the coefficient matrix of the truncated linear system is uniformly bounded. Furthermore, we develop a fast algorithm for solving the corresponding truncated linear system, which preserves the optimal order of the approximate solution with only ${\mathcal{O}}(n\log^{2}n)$ number of multiplications required. Numerical examples complete the paper.     

Iterated fast multiscale Galerkin methods for Fredholm integral equations of second kind with weakly singular kernels

We propose iterated fast multiscale Galerkin methods for the second kind Fredholm integral equations with mildly weakly singular kernel by combining the advantages of fast methods and iteration post-processing methods. To study the super-convergence of these methods, we develop a theoretical framework for iterated fast multiscale schemes, and apply the scheme to integral equations with weakly singular kernels. We show theoretically that even the computational complexity is almost optimal, our schemes improve the accuracy of numerical solutions greatly, and exhibit the global super-convergence. Numerical examples are presented to illustrate the theoretical results and the efficiency of the methods.     

Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind

In this paper we propose new numerical methods for linear Fredholm integral equations of the second kind with weakly singular kernels. The methods are developed by means of the Sinc approximation with smoothing transformations, which is an effective technique against the singularities of the equations. Numerical examples show that the methods achieve exponential convergence, and in this sense the methods improve conventional results where only polynomial convergence have been reported so far.     

Regularity of the solution to a class of weakly singular fredholm integral equations of the second kind

Continuity and differentiability properties of the solution to a class of Fredholm integral equations of the second kind with weakly singular kernel are derived. The equations studied in this paper arise from e.g. potential problems or problems of radiative equilibrium. Under reasonable assumptions it is proved that the solution possesses continuous derivatives in the interior of the interval of integration but may have mild singularities at the end-points.     

Fractional collocation boundary value methods for the second kind Volterra equations with weakly singular kernels

Numerical Algorithms - We discuss the numerical solution to a class of weakly singular Volterra integral equations in this paper. Firstly, the fractional Lagrange interpolation is applied to deal...     

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.     

Numerical methods for 2D weakly singular Volterra integral equations of the second kind

The paper is dedicated to the numerical solution of 2D weakly singular Volterra integral equations with two different kernels. In order to weaken a singularity influence on the numerical computations we transform these equations in both cases into equivalent equations. The piecewise polynomial approximation of the exact solution is then applied. For numerical computing of weakly singular integrals we construct the special Gauss-type cubature formula based on a nonuniform grid. Also we suggest the practical mesh which is less nonuniform than standard graded mesh tk = (k/N)^{r /(1–α)} T, and at the same time it gives an equivalent approximation error for the functions from C^r,α (0, T ]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extrapolation method for solving weakly singular nonlinear Volterra integral equations of the second kind

Based on a new generalization of discrete Gronwall inequality in [L. Tao, H. Yong, A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equality of the second kind, J. Math. Anal. Appl. 282 (2003) 56-62], Navot's quadrature rule for computing integrals with the end point singularity in [I. Navot, A further extension of Euler-Maclaurin summation formula, J. Math. Phys. 41 (1962) 155-184] and a transformation in [P. Baratella, A. Palamara Orsi, A new approach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math. 163 (2004) 401-418], a new quadrature method for solving nonlinear weakly singular Volterra integral equations of the second kind is presented. The convergence of the approximation solution and the asymptotic expansion of the error are proved, so by means of the extrapolation technique we not only obtain a higher accuracy order of the approximation but also get a posteriori estimate of the error.     

Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels

Numerical Algorithms - In this paper, we consider product integration method based on orthogonal polynomials to solve mixed system of Volterra integral equations of the first and second kind with...     

Approximate solution of a class of singular integral equations of second kind

A simple method based on polynomial approximation of a function is employed to obtain approximate solution of a class of singular integral equations of the second kind. For a hypersingular integral equation of the second kind, this method avoids the complex function-theoretic method and produces the known exact solution to Prandtl's integral equation as a special case. For a particular singular integro-differential equation of the second kind, this also produces an approximate solution which compares favourably with numerical results obtained by various Galerkin methods. The convergence of the method for both the equations is also established.     

On the solution of singular integral equations with automorphic kernels by mechanical quadrature

Translated fromIssledovaniya po Prikladnoi Matematike, No. 19, 1992, pp. 111–120.     

On the numerical solution of integral equations with piecewise continuous displacement kernels

Solution of a Fredholm integral equation with a piecewise continuous displacement kernel is considered. It is shown that this problem is equivalent to the solution of an initial value problem for an unusual partial differential equation for continuous functions of two variables. The difference scheme for the numerical solution of the initial value problem is derived. This scheme allows implementation on parallel processors and is of linear complexity. The approach based on the numerical solution of the initial value problem is compared with a corresponding quadrature method and demonstrates certain advantages.This work was supported by the NSF grant DMS-8801961This work was supported by a research grant from the NSERC of Canada     

On approximation and numerical solution of Fredholm integral equations of second kind using quasi-interpolation

In this article a method is presented, which can be used for the numerical treatment of integral equations. Considered is the Fredholm integral equation of second kind with continuous kernel, since this type of integral equation appears in many applications, for example when treating potential problems with integral equation methods.The method is based on the approximation of the integral operator by quasi-interpolating the density function using Gaussian kernels. We show that the approximation of the integral equation, gained with this method, for an appropriate choice of a certain parameter leads to the same numerical results as Nyström’s method with the trapezoidal rule. For this, a convergence analysis is carried out.     

An adaptive method for the numerical solution of Fredholm integral equations of the second kind. I. regular kernels

An adaptive method based on the trapezoidal rule for the numerical solution of Fredholm integral equations of the second kind is developed. The choice of mesh points is made automatically so as to equidistribute both the change in the discrete solution and its gradient. Some numerical experiments with this method are presented.