Ahmed Zidna

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.

A general family of third order method for finding multiple roots

Abstract In this paper, we describe a general family of iterative methods for approximating a multiple root z with multiplicity m of a complex defined function. Almost of the family of the methods existing in the literature that use two-function and one-derivative evaluations are a special choice of this general method. We give some conditions to have the third order of convergence and we discuss how to choose a small asymptotic error constant which may be affect the speed of the convergence. Using Mathematica with its high precision compatibility, we present some numerical examples to confirm the theoretical results.

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.

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.

Construction of flexible blending parametric surfaces via curves

The main purpose of this paper is to provide a method that allows to solve the blending problem of two parametric surfaces. The blending surface is constructed with a collection of space curves defined by point pairs on the blending boundaries of given primary surfaces. Bezier and C-cubic curves are used to interpolate the blending boundaries. The blending surface is G^n continuously connected to the primary surfaces.

A two-steps algorithm for approximating real roots of a polynomial in Bernstein basis

The surface/curve intersection problem, through the resultants process results in a high degree (n>=100) polynomial equation on [0,1] in the Bernstein basis. The knowledge of multiplicities of the roots is critical for the topological coherence of the results. In this aim, we propose an original two-steps algorithm based on successive differentiations which separates any root (even multiple) and guarantees that the assumptions of Newton global convergence theorem are satisfied. The complexity is @q(n^4) but the algorithm can easily be parallelized. Experimental results show its efficiency when facing ill-conditioned polynomials.

Recursive de Casteljau bisection and rounding errors

Rounding errors of the de Casteljau bisection algorithm applied recursively to finding zeros of polynomials of one or more variables are analyzed. Apart from error bounds for this procedure, the paper contains a proof of the so called numerical variation diminishing property (formulated for one-dimensional case), which is significant in practical procedures of solving algebraic equations.

Two cubic spline methods for solving Fredholm integral equations

Abstract In this work, we propose two methods based on the use of natural and quasi cubic spline interpolations for approximating the solution of the second kind Fredholm integral equations. Convergence analysis is established. Some numerical examples are given to show the validity of the presented methods.

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.