A solution method for hypersingular integral equations

The Neumann problem for the Helmholtz equation is considered. The double-layer potential is used to reduce the problem to a hypersingular integral equation. The properties of the hypersingular operator in a neighborhood lead to a method for approximate solution of the hypersingular equation with an arbitrary contour. Some numerical results are reported.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 130–136, 1993.     

Newton methods for a class of nonlinear hypersingular integral equations

Different iterative schemes based on collocation methods have been well studied and widely applied to the numerical solution of nonlinear hypersingular integral equations (Capobianco et al. 2005). In this paper we apply Newton’s method and its modified version to solve the equations obtained by applying a collocation method to a nonlinear hypersingular integral equation of Prandtl’s type. The corresponding convergence results are derived in suitable Sobolev spaces. Some numerical tests are also presented to validate the theoretical results.     

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.     

Approximate solution for a class of two-dimensional nonlinear singular integral equations by the contraction mapping method

We obtain solutions for a class of two-dimensional nonlinear singular integral equations with Hilbert kernel using the contraction mapping method and find the rate of convergence of successive approximations to the exact solution.     

二维Fredholm方程的Taylor展开式解法

本文讨论了一类二维Fredholm方程的一种近拟解,通过利用二元函数的Taylor展开式,积分方程转化成一个关于未知函数及其相应的偏导数的线性代数方程组.数值例子表明了该方法的有效性.     

Hierarchical basis methods for hypersingular integral equations

In this paper we present hierarchical basis methods for theGalerkin approximation of hypersingular integral equations onthe interval = (–1,1). The condition number of the stiffnessmatrix with respect to the hierarchical basis is shown to behavelike O(|logh|²). The implementations are based on the preconditionedconjugate gradient method using a hierarchical basis (HB) preconditioner.The numerical results are presented with a comparison betweenthe HB preconditioner and the BPX (Bramble, Pasciak and Xu)preconditioner.     

An Approximate solution for a class of non-linear hill's equations

The object of this paper is to prove the existence of an approximate solution in the mean for some non-hear differential equations. Further, we investigate the behavior of the class or solutions which may be associated with the differential equalion.     

On the solution of a class of integral equations

In this paper we consider a class of Fredholm integral equations of the first kind which arise in a large number of problems in applied mathematics. Although only certain special cases of the equations can be solved exactly, it is shown that a constructive method can be developed for reformulating the equations as Fredholm integral equations of the second kind. This approach will be seen to cover and bring together the large number of isolated cases of the equations which have appeared in the literature. Several examples are given to illustrate the general method.     

A Nitsche-based domain decomposition method for hypersingular integral equations

We introduce and analyze a Nitsche-based domain decomposition method for the solution of hypersingular integral equations. This method allows for discretizations with non-matching grids without the necessity of a Lagrangian multiplier, as opposed to the traditional mortar method. We prove its almost quasi-optimal convergence and underline the theory by a numerical experiment.     

On the solution of a class of integral equations II

A class of integral equations is investigated, particular examples of which occur in the consideration of certain three- and four-part mixed boundary-value problems in applied mathematics. A constructive method is given for reformulating the integral equations as Fredholm integral equations of the second kind and three examples are examined in detail to illustrate the general methods developed in the paper.     

ON DIRECT METHOD OF SOLUTION FOR A CLASS OF SINGULAR INTEGRAL EQUATIONS

In this article, by introducing characteristic singular integral operator and associate singular integral equations (SIEs), the authors discuss the direct method of solution for a class of singular integral equations with certain analytic inputs. They obtain both the conditions of solvability and the solutions in closed form. It is noteworthy that the method is different from the classical one that is due to Lu.     

Approximate solution of boundary integral equations for biharmonic problems in non-smooth domains

This paper deals with approximate solutions to integral equations arising in boundary value problems for the biharmonic equation in simply connected piecewise smooth domains. The approximation method considered demonstrates excellent convergence even in the case of boundary conditions discontinuous at corner points. In an application we obtain very accurate approximations for some characteristics of two-dimensional Stokes flow in non-smooth domains. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Preconditined semilinear stochastic singular and hypersingular integral equations

In this paper, three classes of preconditioners are proposed for solving some stochastic integral equations with the weakly singular kernel and the hypersingular kernel. The first and the second class of preconditioners are based on circulant operators, but the third class of preconditioners is based on iterative substructuring. It is proved that substructuring preconditioners can be better than other preconditioners. Also, the spaces of solutions are discussed such that the solutions of these equations are smooth, therefore, we give special Banach spaces for these integral equations. Finally, numerical results which support our theories are presented     

Adaptive meshless Galerkin boundary node methods for hypersingular integral equations

Adaptive refinement techniques are developed in this paper for the meshless Galerkin boundary node method for hypersingular boundary integral equations. Two types of error estimators are derived. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two consecutive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the numerical solution itself and its projection. These error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization scheme is presented to accomodate the non-local property of hypersingular integral operators for the needed computable local error indicators. The convergence of the adaptive meshless techniques is verified theoretically. To confirm the theoretical results and to show the efficiency of the adaptive techniques, numerical examples in 2D and 3D with high singularities are provided.     

Approximate solution of integral equations and convolution integrals using Legendre polynomials

It has been argued that Chebyshev polynomials are ideal to use as approximating functions to obtain solutions of integral equations and convolution integrals on account of their fast convergence. Using the standard deviation as a measure of the accuracy of the approximation and the CPU time as a measure of the speed, we find that for reasonable accuracy Legendre polynomials are more efficient.     

A method for accelerating the iterative solution of a class of Fredholm integral equations

The present paper extends the synthetic method of transport theory to a large class of integral equations. Convergence and divergence properties of the algorithm are studied analytically, and numerical examples are presented which demonstrate the expected theoretical behavior. It is shown that, in some instances, the computational advantage over the familiar Neumann approach is substantial.This authors acknowledge with pleasure conversations with Paul Nelson. Thanks are due also to Janet E. Wing, whose computer program was used in making the calculations reported in Section 8.This work was performed in part under the auspices of USERDA at the Los Alamos Scientific Laboratory of the University of California, Los Alamos, New Mexico.