Quadrature methods for integro-differential equations of Prandtl's type in weighted spaces of continuous functions

The paper deals with the approximate solution of integro-differential equations of Prandtl's type. Quadrature methods involving ``optimal'' Lagrange interpolation processes are proposed and conditions under which they are stable and convergent in suitable weighted spaces of continuous functions are proved. The efficiency of the method has been tested by some numerical experiments, some of them including comparisons with other numerical procedures. In particular, as an application, we have implemented the method for solving Prandtl's equation governing the circulation air flow along the contour of a plane wing profile, in the case of elliptic or rectangular wing-shape.Comment: 34 page

The numerical solution of Cauchy singular integral equations with additional fixed singularities

In this paper we propose a quadrature method for the numerical solution of Cauchy singular integral equations with additional fixed singularities. The unknown function is approximated by a weighted polynomial which is the solution of a finite dimensional equation obtained discretizing the involved integral operators by means of a Gauss-Jacobi quadrature rule. Stability and convergence results for the proposed procedure are proved. Moreover, we prove that the linear systems one has to solve, in order to determine the unknown coefficients of the approximate solutions, are well conditioned. The efficiency of the proposed method is shown through some numerical examples

Approximation of Hilbert and Hadamard transforms on (0, +∞)

The authors propose a numerical method for computing Hilbert and Hadamard transforms on (0, +∞) by a simultaneous approximation process involving a suitable Lagrange polynomial of degree s and “truncated” Gaussian rule of order m, with s&lt

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

Remarks on two integral operators and numerical methods for CSIE

In this paper the author extends the mapping properties of some singular integral operators in Zygmund spaces equipped with uniform norm. As by-product quadrature methods for solving CSIE having index $0$ and $1$ are proposed. Their stability and convergence are proved and error estimates in Zygmund norm are given. Some numerical tests are also shown

Nyström method for systems of integral equations on the real semiaxis

In this paper, the authors introduce a Nystrom method for solving systems of Fredholm integral equations on the real semiaxis. They prove that the method is stable and convergent. Moreover, they show some numerical tests that confirm the error estimates. Finally, they discuss a specific application to an inverse scattering problem for the Schrodinger equation