Chafik Allouch

Solving Fredholm integral equations by approximating kernels by spline quasi-interpolants

We study two methods for solving a univariate Fredholm integral equation of the second kind, based on (left and right) partial approximations of the kernel K by a discrete quartic spline quasi-interpolant. The principle of each method is to approximate the kernel with respect to one variable, the other remaining free. This leads to an approximation of K by a degenerate kernel. We give error estimates for smooth functions, and we show that the method based on the left (resp. right) approximation of the kernel has an approximation order O(h5) (resp. O(h6)). We also compare the obtained formulae with projection methods.

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.

A collocation method for the numerical solution of a two dimensional integral equation using a quadratic spline quasi-interpolant

In this paper, we propose an interesting method for approximating the solution of a two dimensional second kind equation with a smooth kernel using a bivariate quadratic spline quasi-interpolant (abbr. QI) defined on a uniform criss-cross triangulation of a bounded rectangle. We study the approximation errors of this method together with its Sloan’s iterated version and we illustrate the theoretical results by some numerical examples.

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.

Original article: Iteration methods for Fredholm integral equations of the second kind based on spline quasi-interpolants

In this paper, we propose an efficient iteration algorithm for Fredholm integral equations of the second kind based on spline quasi-interpolants (abbr. QIs). We show that for every iteration step we obtain superconvergence rates. A superconvergent method called functional approximation method based on QIs is also developed. We illustrate our results by numerical experiments.

Original article: A modified Kulkarni's method based on a discrete spline quasi-interpolant

We use a discrete spline quasi-interpolant (abbr. dQI) defined on a bounded interval for the numerical solution of linear Fredholm integral equations of the second kind with a smooth kernel by collocation and a modified Kulkarnis method together with its Sloans iterated version. We study the approximation errors of these methods and we illustrate the theoretical results by a numerical example.

Collocation methods for solving multivariable integral equations of the second kind

In a recent paper (Allouch, in press) [5] on one dimensional integral equations of the second kind, we have introduced new collocation methods. These methods are based on an interpolatory projection at Gauss points onto a space of discontinuous piecewise polynomials of degree r which are inspired by Kulkarnis methods (Kulkarni, 2003) [10], and have been shown to give a 4r+4 convergence for suitable smooth kernels. In this paper, these methods are extended to multi-dimensional second kind equations and are shown to have a convergence of order 2r+4. The size of the systems of equations that must be solved in implementing these methods remains the same as for Kulkarnis methods. A two-grid iteration convergent method for solving the system of equations based on these new methods is also defined.

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.