Maria Grazia Russo

Numerical methods for Fredholm integral equations on the square

Abstract In this paper we shall investigate the numerical solution of two-dimensional Fredholm integral equations by Nystrom and collocation methods based on the zeros of Jacobi orthogonal polynomials. The convergence, stability and well conditioning of the method are proved in suitable weighted spaces of functions. Some numerical examples illustrate the efficiency of the methods.

Numerical methods for Fredholm integral equations with singular right-hand sides

Fredholm integral equations with the right-hand side having singularities at the endpoints are considered. The singularities are moved into the kernel that is subsequently regularized by a suitable one-to-one map. The Nyström method is applied to the regularized equation. The convergence, stability and well conditioning of the method is proved in spaces of weighted continuous functions. The special case of the weakly singular and symmetric kernel is also investigated. Several numerical tests are included.

Nyström methods for bivariate Fredholm integral equations on unbounded domains

Abstract In this paper we propose a numerical procedure in order to approximate the solution of two-dimensional Fredholm integral equations on unbounded domains like strips, half-planes or the whole real plane. We consider global methods of Nystrom types, which are based on the zeros of suitable orthogonal polynomials. One of the main interesting aspects of our procedures regards the “quality” of the involved functions, since we can successfully manage functions which are singular on the finite boundaries and can have an exponential growth on the infinite boundaries of the domains. Moreover the errors of the methods are comparable with the error of best polynomial approximation in the weighted spaces of functions that we go to treat. The convergence and the stability of the methods and the well conditioning of the final linear systems are proved and some numerical tests, which confirm the theoretical estimates, are given.

Extended Lagrange interpolation in L1 spaces

Let w(x)=e−xβxα, w¯(x)=xw(x) and denote by {pm(w)}m,{pn(w¯)}n the corresponding sequences of orthonormal polynomials. The zeros of the polynomial Q2m+1=pm+1(w)pm(w¯) are simple and are sufficiently far among them. Therefore it is possible to construct an interpolation process essentially based on the zeros of Q2m+1, which is called ”Extended Lagrange Interpolation”. Here we study the convergence of this interpolation process in suitable weighted L1 spaces. This study completes the results given by the authors in previous papers in weighted Lup((0,+∞)), for 1≤p≤∞. Moreover an application of the proposed interpolation process in order to construct an e cient product quadrature scheme for weakly singular integrals is given.