M. C. De Bonis

Approximation of the Hilbert Transform on the real semiaxis using Laguerre zeros

The authors propose two new algorithms for the computation of Cauchy principal value integrals on the real semiaxis. The proposed quadrature rules use zeros of Laguerre polynomials. Theoretical error estimates are proved and some numerical examples are showed.

Approximation of the hilbert transform on the real line using Hermite zeros

The authors study the Hilbert Transform on the real line. They introduce some polynomial approximations and some algorithms for its numerical evaluation. Error estimates in uniform norm are given.

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.

Gaussian quadrature rules with exponential weights on (−1, 1)

We study the behavior of some “truncated” Gaussian rules based on the zeros of Pollaczek-type polynomials. These formulas are stable and converge with the order of the best polynomial approximation in suitable function spaces. Moreover, we apply these results to the related Lagrange interpolation process and to prove the stability and the convergence of a Nyström method for Fredholm integral equations of the second kind. Finally, some numerical examples are shown.

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

Numerical methods for some special Fredholm integral equations on the real line

The authors propose a simple numerical method to approximate the solution of CSIE. The convergence and the stability of the procedure are proved and some numerical examples are shown.

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.

Numerical treatment of a class of systems of Fredholm integral equations on the real line

In this paper the authors propose a Nystrom method based on a “truncated” Gaussian rule to solve systems of Fredholm integral equations on the real line. They prove that it is stable and convergent and that the matrices of the solved linear systems are well-conditioned. Moreover, they give error estimates in weighted uniform norm and show some numerical tests. Fredholm integral equations, Nystrom method, truncated Gaussian rule

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.