C. Laurita

Numerical solution of systems of Cauchy singular integral equations with constant coefficients

Abstract This paper deals with the numerical solution of a class of systems of Cauchy singular integral equations with constant coefficients. The proposed procedure consists of two basic steps: the first one is to consider a modified problem equivalent to the original one under suitable conditions, the second one is to approximate its solution by means of a vector of polynomial functions. Such array is constructed by applying a quadrature type method, based on Gaussian rules, that leads to solve a determined and well conditioned linear system. The convergence and stability of the method are proved in weighted L 2 spaces. Some numerical tests are also shown.

A Nyström method for a boundary integral equation related to the Dirichlet problem on domains with corners

The authors consider the interior Dirichlet problem for Laplace’s equation on planar domains with corners. They provide a complete analysis of a natural method of Nyström type based on the global Gauss–Lobatto quadrature rule, in order to approximate the solution of the corresponding double layer boundary integral equation. Mellin-type integral operators are involved and, as usual, a modification of the method close to the corners is needed. A new modification is proposed and the convergence and stability of the “modified” quadrature method are proved. Some numerical tests are also included.

Nyström method for Cauchy singular integral equations with negative index

In this paper, the authors propose a Nystrom method to approximate the solutions of Cauchy singular integral equations with constant coefficients having a negative index. They consider the equations in spaces of continuous functions with weighted uniform norm. They prove the stability and the convergence of the method and show some numerical tests that confirm the error estimates.

A modified Nyström method for integral equations with Mellin type kernels

The aim of this paper is to propose a new modified Nystrom method for the approximation of the solutions of second kind integral equations with fixed singularities of Mellin convolution type. The stability and the convergence are proved in L 2 spaces and error estimates in L 2 norm are given. Finally, numerical tests showing the effectiveness of the method are presented.

A quadrature method for systems of Cauchy singular integral equations

The aim of this paper is to propose a numerical method approximating the solutions of a system of CSIE. The stability and the convergence of the method are proved in weighted L2 spaces. An application to the numerical resolution of CSIE on curves is also given. Finally some numerical tests confirming the error estimates are shown. Cauchy singular integral equation, quadrature method, Lagrange interpolation 65R20, 45E05

A Nystrm method for integral equations with fixed singularities of Mellin type in weighted Lp spaces

We consider integral equations of the second kind with fixed singularities of Mellin type. According to the behavior of the Mellin kernel, we first determine suitable weighted Lp spaces where we look for the solution. Then, for its approximation, we propose a numerical method of Nystrm type based on a GaussJacobi quadratura formula. Actually, a slight modification of the classical procedure is introduced in order to prove convergence results in weighted Lp spaces. Moreover, a preconditioning technique allows us to solve well conditioned linear systems. We show the efficiency of the proposed method through some numerical tests.

On the stability of a modified Nyström method for Mellin convolution equations in weighted spaces

This paper deals with the numerical solution of second kind integral equations with fixed singularities of Mellin convolution type. The main difficulty in solving such equations is the proof of the stability of the chosen numerical method, being the noncompactness of the Mellin integral operator the chief theoretical barrier. Here, we propose a Nyström method suitably modified in order to achieve the theoretical stability under proper assumptions on the Mellin kernel. We also provide an error estimate in weighted uniform norm and prove the well-conditioning of the involved linear systems. Some numerical tests which confirm the efficiency of the method are shown.