M. Tahrichi

Product integration methods based on discrete spline quasi-interpolants and application to weakly singular integral equations

Quadrature formulae are established for product integration rules based on discrete spline quasi-interpolants on a bounded interval. The integrand considered may have algebraic or logarithmic singularities. These formulae are then applied to the numerical solution of integral equations with weakly singular kernels.

Superconvergent Nyström and degenerate kernel methods for Hammerstein integral equations

In a recent paper, we introduced new methods called superconvergent Nystrom and degenerate kernel methods for approximating the solution of Fredholm integral equations of the second kind with a smooth kernel. In this paper, these methods are applied to numerically solve the Hammerstein equations. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree @?r-1, we prove that, as for Fredholm integral equations, the proposed methods exhibit convergence orders 3r and 4r for the iterated version. Several numerical examples are given to demonstrate the effectiveness of the current methods.

A simple method for constructing integro spline quasi-interpolants

In this paper, we study a new method for function approximation from the given integral values over successive subintervals by using cubic B-splines. The method does not need any additional end conditions and it is easy to be implemented without solving any system of linear equations. The method is able to approximate the original function and its first and second-order derivatives over the global interval successfully. The approximation errors are well studied. Numerical results illustrate that our method is very effective.

Chordal cubic spline quasi interpolation

This paper studies cubic spline quasi-interpolation of parametric curves through sequences of points in any space dimension. We show that if the parameter values are chosen by chord length, the order of accuracy is four. We also use this chordal cubic spline quasi interpolant to approximate the arc length derivatives and the length of the parametric curve.

Numerical solutions of weakly singular Hammerstein integral equations

In this paper, several methods for approximating the solution of Hammerstein equations with weakly singular kernels are considered. The paper is motivated by the results reported in papers [7, 12]. The orders of convergence of the proposed methods and those of superconvergence of the iterated methods are analyzed. Numerical examples are given to illustrate the theoretical results.

Discrete superconvergent Nyström method for integral equations and eigenvalue problems

In this paper, discrete superconvergent Nystrom method is studied for solving the second kind Fredholm integral equations and eigenvalue problems of a compact integral operator with a smooth kernel. We use interpolatory projections at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ? r - 1 . We analyze the convergence of this method and its iterated version and we establish superconvergence results. Numerical examples are presented to illustrate the obtained theoretical estimates.

Superconvergent spline quasi-interpolants and an application to numerical integration

In this paper, we present a new technique to get superconvergence phenomenon of spline quasi-interpolants at the knots of the partition. This method gives rise to good approximation not only at these knots but also on the whole domain of definition. Moreover, we give an application to numerical integration. Numerical results are given to illustrate the theoretical ones.