Abdelleh Lamnii

Sextic spline solution of fifth-order boundary value problems

There are few techniques to numerically solve fifth-order boundary-value problems (BVPs). In this paper two sextic spline collocation methods are developed and analyzed. The first one uses spline interpolants and the second is based on spline quasi-interpolants. They are both proved to be second-order convergent. Numerical results verify the order of convergence predicted by the analysis.

Spline collocation method for solving linear sixth-order boundary-value problems

There are few techniques to numerically solve sixth-order boundary-value problems. In this paper, septic-spline collocation method based on spline interpolants is developed and analysed. There is proved to be second-order convergent. Numerical results verify the order of convergence predicted by the analysis. We also give a comparison of this with those developed by El-Gamel, Cannon and Zayed in 2003 and Akram and Siddiqi in 2006.

Sextic spline collocation methods for nonlinear fifth-order boundary value problems

In this paper, two sextic-spline collocation methods are developed and analysed for approximating solutions of nonlinear fifth-order boundary-value problems. The first method uses a spline interpolant and the second one is based on a spline quasi-interpolant, which are constructed from sextic splines. They are both proved to be second-order convergent. Numerical results confirm the order of convergence predicted by the analysis. It has been observed that the methods developed in this paper are better than the others given in the literature.

A normalized basis for condensed C 1 Powell-Sabin-12 splines

The purpose of this article is the construction of a normalized basis for a quadratic condensed Powell-Sabin-12 macro-element space introduced by Alfeld et al. (2010). The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction of this basis is adopted from Dierckx (1997) and Speleers (2010a), and is based on the determination of a set of triangles that must contain a specific set of points. The proposed basis can only be constructed on triangulations with a maximal angle less than π 2 . B-spline basis of the quadratic condensed Powell-Sabin-12 spline space.Determination of control points.Discrete and differentiable quasi-interpolants.

Original article: Uniform tension algebraic trigonometric spline wavelets of class C2 and order four

In this paper we first describe a multiresolution curve representation based on periodic uniform tension algebraic trigonometric (UTAT) spline wavelets of class C^2 and order four. Then we determine the decomposition and the reconstruction vectors corresponding to UTAT-spline spaces. Finally, we give some applications in order to illustrate the efficiency of the proposed approach.

Construction of spherical spline quasi-interpolants based on blossoming

A general theory of quasi-interpolants based on quadratic spherical Powell-Sabin splines on spherical triangulations of a sphere-like surface S is developed by using polar forms. As application, various families of discrete and differential quasi-interpolants reproducing quadratic spherical Bezier-Bernstein polynomials or the whole space of the spherical Powell-Sabin quadratic splines of class C^1 are presented.

Multiresolution Analysis of UTAT B‐spline Curves

In this paper, we describe a multiresolution curve representation based on periodic uniform tension algebraic trigonometric (UTAT) spline wavelets of class ??? and order four. Then we determine the decomposition and the reconstruction vectors corresponding to UTAT‐spline spaces. Finally, we give some applications in order to illustrate the efficiency of the proposed approach.

Computation of Hermite interpolation in terms of B-spline basis using polar forms

The aim of this paper is to present an Hermite interpolation problem with B-splines of high degree of smoothness. More precisely, we use polar forms to find the B-spline control points for Hermite interpolation. The resulting formula is used to give Hermite basis functions. In particular, Quadratic C 1 and cubic C 2 interpolations with sharp parameters are analyzed.

Constructing B-spline representation of quadratic Sibson-Thomson splines

In this paper, we show how to construct a normalized B-spline basis for a special C 1 continuous splines of degree 2, defined on Sibson-Thomson refinement. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The dilatation equation can be found by applying the dyadic subdivision scheme directly to the Sibson-Thomson spline basis functions. As an application, a quasi-interpolation method, based on this Sibson-Thomson B-spline representation, is described which can be used for the efficient visualization of gridded surface data. B-spline basis of Sibson-Thomson splines.Refinement equation.Quasi-interpolants.

Algebraic hyperbolic spline quasi-interpolants and applications

Abstract In this paper, a construction of Marsden’s identity for UAH B-splines (i.e. Uniform Algebraic Hyperbolic B-splines) is developed and a clear proof is given. With the help of this identity, quasi-interpolant schemes that produce the space of algebraic hyperbolic functions are derived. Efficient quadrature rules, based on integrating some of these quasi-interpolant schemes, are constructed and studied. Numerical results that illustrate the effectiveness of these rules are presented.