Hamid Mraoui

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.

A normalized basis for C1 cubic super spline space on Powell-Sabin triangulation

In this paper, we describe the construction of a suitable normalized B-spline representation for bivariate C^1 cubic super splines defined on triangulations with a Powell-Sabin refinement. The basis functions have local supports, they form a convex partition of unity, and every spline is locally controllable by means of control triangles. As application, discrete and differential quasi-interpolants of optimal approximation order are developed and numerical tests for illustrating theoretical results are presented.

Computing the range of values of real functions using B-spline form

Abstract In this paper, we present an easy and efficient method for computing the range of a function by using spline quasi-interpolation. We exploit the close relationship between the spline function and its control polygon and use tight subdivision technique in order to obtain monotonic splines which make the range of the spline easy to compute. The proposed method is useful in case of given scattered data generated by some (unknown) function f or scientific measurements. Several numerical examples are given, for cubic and quintic quasi-interpolant approximant, to illustrate the efficiency and the performance of our method.

Construction of quintic Powell–Sabin spline quasi-interpolants based on blossoming

Abstract By blossoming Marsden’s identity, we investigate local quasi-interpolation schemes for C 2 -continuous quintic Powell–Sabin splines represented with a normalized B-spline basis. As applications, various families of discrete and differential quasi-interpolants reproducing quintic polynomials are presented.

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.