# Abedallah M RababahUnited Arab Emirates University | UAEU · Department of Mathematical Sciences

Abedallah M Rababah

Professor (Full) of Mathematics

## About

59

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655

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Introduction

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September 2012 - August 2015

July 2009 - August 2012

July 2009 - August 2012

## Publications

Publications (59)

In this paper, an approximation procedure for space curves is described, which significantly improves the standard approximation rate via parametric Taylor's approximations. The method takes advantages of the freedom in the choice of the parametrization and yields the order (m + 1) + [(m + 1)(2d - 1)] for a curve in Rd, where m is the degree of the...

We describe an approximation method for planar curves that significantly improves the standard rate obtained by local Taylor approximations. The method exploits the freedom in the choice of the parametrization and achieves the order 4m/3 where m is the degree of the approximating polynomial parametrization.

The problem of degree reduction and degree raising of triangular Bézier surfaces is considered. The L2 and l2 measures of distance combined with the least-squares method are used to get a formula for the Bézier points. The methods use the matrix representations of the degree reduction and degree raising.

In this article, it is shown that a space curve in R-d can be approximated by a piecewise polynomial curve of degree m with order (m + 1) + [(m + 1)/(2d - 1)] rather than m + 1. Moreover, we show that the optimal order (m + 1) + [(m - 1)/(d - 1)] is possible for a particular set of curves of nonzero measure. Analogous results were shown to be true...

This paper presents decomposition of the fourth-order Euler-type linear time-varying system (LTVS) as a commutative pair of two second-order Euler-type systems. All necessary and sufficient conditions for the decomposition are deployed to investigate the commutativity, sensitivity, and the effect of disturbance on the fourth-order LTVS. Some system...

In this paper, the Chebyshev-I conformable differential equation is considered. A proper power series is examined; there are two solutions, the even solution and the odd solution. The Rodrigues’ type formula is also allocated for the conformable Chebyshev-I polynomials.

This paper considers Chebyshev weighted G3-multi-degree reduction of B´ezier curves. Exact degree reduction is not possible, based on this fact, approximative process to reduce a given B´ezier curve of higher degree n to a B´ezier curve of lower degree m, m<n is required. The weight function w[t]=2t(1−t),t∈[0,1] is used with the L2-norm in multi de...

In this paper, we are concerned about numerical solutions to ODEs and PDEs that are used to model and describe real-life problems. Usually, these equations are approximated numerically because it is convenient and on-hand to be calculated on computing devices. The Burger-Huxley partial differential equations model the interaction between reactions,...

In this article, the best uniform approximation for the hyperbola of degree 6 that has approximation order 12 is found. The associated error function vanishes 12 times and equioscillates 13 times. For an arc of the hyperbola, the error is bounded by 2:4 x 10-4. We explain the details of the derivation and show how to apply the method. The method is...

In this paper, a new method for the approximation of offset curves is presented using the idea of the parallel derivative curves. The best uniform approximation of degree 3 with order 6 is used to construct a method to find the approximation of the offset curves for Bezier curves. The proposed method is based on the best uniform approximation, and...

Over the years, switches and network routers have been compromised frequently, and a lot of vulnerabilities have occurred in the network infrastructure. Secure routing (SR) is one of the challenges that currently exists in computer networks. Software-defined networks (SDN) are designed by assuming that routers or switches are trustworthy. In SDN, u...

In this paper, the behavior of the Hermite-Fej´er interpolation for functionswithderivativesofboundedvariationon[−1,1]isstudiedbytakingtheinterpolation over the zeros of Chebyshev polynomials of the second kind. An estimate for the rate of convergence using the zeros of the Chebyshev polynomials of the second kind is given.

In the rural side, due to the absence of cardiovascular ailment centers, around 12 million people passing away worldwide reported by WHO. The principal purpose of coronary illness is a propensity of smoking. Our Cluster based disease Diagnosis (CDD) applies the ML classifiers to improve the prediction accuracy of cardiovascular diseases. For this w...

p>Mathematically, circles are represented by trigonometric parametric equations and implicit equations. Both forms are not proper for computer applications and CAD systems. In this paper, a quintic polynomial approximation for a circular arc is presented. This approximation is set so that the error function is of degree $10$ rather than $6$; the Ch...

In this paper, a method to approximate curves by polynomials of degree nine is presented. The resulting approximation has order eighteen. The method is applied to approximate a circular arc, and the error function is studied and characterized, and its extrema and zeros are derived.

In this paper, the change of bases transformations between the Bernstein polynomial basis and the Chebyshev polynomial basis of the fourth kind are studied and the matrices of transformation among these bases are constructed. Some examples are given.

In this article, a quadrature formula of degree 2 is given that has degree of exactness 3 and order 5. The formula is valid for any planar curve given in parametric form unlike existing Gaussian quadrature formulas that are valid only for functions.

We consider the weighted-multi-degree reduction of Bézier curves. Based on the fact that exact degree reduction is not possible, therefore approximative process to reduce a given Bézier curve of high degree n to a Bézier curve of lower degree m, \(m<n\) is needed. The weight function is used to better representing the approximative curve at some pa...

In this paper, the Hermite-Fejér interpolations are studied over the roots of some classes of orthogonal polynomials. In particular, the Hermite-Fejér interpolations to functions of bounded variation and functions with derivatives of bounded variations are investigated. New related results and open problems are stated.

In this paper, a quadratic spline is presented. The boundary conditions are adjusted to get new parameters that are used to get better approximation. Some numerical results are given to demonstrate the advantages and efficiency of the method. The proposed method is more accurate than the traditional quadratic method.

A uniform quadratic approximation of degree 2 is created in explicit parametric form to represent elliptical arcs. The error function is identical to that of the Chebyshev polynomial of degree 4 and equioscillates five times with an approximation order of four. In this paper we provide the approximation method, show it is efficient, its error bound...

In this article, the issue of the best uniform approximation of circular arc with parametrically defined polynomial curves is considered. The best uniform approximation of degree 5 to a circular arc is given in explicit form.
The approximation is constructed so that the error function is the monic Chebyshev polynomial of degree 10; the error functi...

In this paper, a new approach for multi-degree reduction of Said-Ball curves is investigated. Conditions for continuities and tangent continuities at both boundaries of the curve are given. The distance between the original Said-Ball curve and the degree reduced Said-Ball curve is measured in L2-norm under the satisfaction of G°-and G¹-continuity c...

A high accuracy quartic approximation for circular arc is given in this article. The approximation is constructed so that the error function is of degree 8 with the least deviation from the x-axis; the error function equioscillates 9 times; the approximation order is 8. The numerical examples demonstrate the efficiency and simplicity of the approxi...

In this paper, weighted G
0-and G
1-multi-degree reduction of Bézier curves are considered. The degree reduction of a given Bézier curve of degree n is used to write it as a Bézier curve of degree m, m < n. Exact degree reduction is not possible, and, therefore approximation methods are used. The weight function w[t] = 2t(1 − t), t ∈ [0, 1] is used...

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In this paper, the basis transformation between the Chebyshev polynomials of third kind and the Bernstein polynomials is considered and the transformation matrices are derived.

A disk Bézier curve is a Bézier curve whose control points are disks. It can be considered as a parametric curve with error tolerances. In this paper, we propose method to find G
2-multi degree reduction of disk Bézier curves based on two stages. The center and the radius curves are degree reduced using G
2-continuity in the first stage and minimiz...

In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P n , r ( u , v , w ) , r = 0 , 1 , … , n , n ≥ 0 on the triangular domain T = { ( u , v , w ) : u , v , w ≥ 0 , u + v + w = 1 } are constructed, where u , v , w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very p...

In this article, the issue of the best uniform approximation of circular arcs with parametrically defined polynomial curves is considered. The best uniform approximation of degree 2 to a circular arc is given in explicit form. The approximation is constructed so that the error function is the Chebyshev polynomial of degree 4; the error function equ...

In this paper, weighted G1-multi-degree reduction of B´ezier curves is considered. The degree reduction of a given B´ezier curve of degree n is used to write it as a B´ezier curve of degree m,m < n. Exact degree reduction is not possible, and, therefore, approximation methods are used. The weight function w(t) = 2t(1 - t), t [0, 1] is used with the...

In this paper, we propose methods to find a G k -multi-degree reduction of disk Bézier curves for k = 0 , 1 . The methods are based on degree reducing the center and radius curves using G k -continuity and minimizing the corresponding errors. Some examples and comparisons are given to illustrate the efficiency and simplicity of the proposed methods...

In this paper, we propose methods based on two stages to find G
0- and G
1- multi degree reductions of disk Bezier curves. The first stage is degree reducing the centre and radius curves and the second stage is approximating the error radius curve. An example is given to illustrate the efficiency of the proposed method which shows that our method o...

In this paper, the issue of multi-degree reduction of Bézier curves with C1 and G2-continuity at the end points of the curve is considered. An iterative method, which is the first of this type, is derived. It is shown that this algorithm converges and can be applied iteratively to get the required accuracy. Some examples and figures are given to de...

In the paper [A. Rababah, M. Alqudah, Jacobi-weighted orthogonal polynomials on triangular domains, J. Appl. Math. 3 (2005) 205-217.], Jacobi-weighted orthogonal polynomials P"n","r^(^@a^,^@b^,^@c^)(u,v,w),@a,@b,@c>-1 on the triangular domain T for values of @a,@b,@c>-1 in the plane @a+@b+@c=0 are constructed. In this paper, the results are general...

In this paper, we consider the issue of dual functions for the Bernstein basis which satisfy boundary conditions. The Jacobi weight function with the usual inner product in the Hilbert space are used. Some examples of the transformation matrices are given. Some figures for the weighted dual functions of the Bernstein basis with respect to the Jacob...

In this paper, we give a new, simple, and efficient method for evaluating the pth derivative of the Jacobi polynomial of degree n. The Jacobi polynomial is written in terms of the Bernstein basis, and then the pth derivative is obtained. The results are given in terms of both Bernstein basis of degree n − p and Jacobi basis form of degree n − p and...

In this paper, bounds on the sum of the fundamental polynomials Hkn(cos θ) associated with the Hermite-Fejer interpolation on the roots xkn = cos θkn of the Jacobi Polynomials P (α,β) n (cos θ) ,α , β >−1 are presented. n (x) ,α , β >−1 are orthogonal polynomials on (−1,1) with respect to the weight function w(x )=( 1− x)α(1+ x)β ,α , β >−1. The Ja...

In this article, we find the optimal r times degree reduction of Bézier curves with respect to the Jacobi-weighted L 2-norm on the interval [0, 1]. This method describes a simple and efficient algorithm based on matrix computations. Also, our method includes many previous results for the best approximation with L 1, L 2, and L ∞-norms. We give some...

We find an explicit formula for the weighted dual functions of the Bernstein polynomials with respect to the Jacobi weight function using the usual inner product in the Hilbert space L2[0,1]. We define the weighted dual functionals of the Bernstein polynomials, which are used to find the coefficients in the least squares approximation.

This paper presents methods to compute integrals of the Jacobi polynomials
by the representation in terms of the Bernstein — B´ezier basis. We do this
because the integration of the Bernstein — B´ezier form simply corresponds to applying
the de Casteljau algorithm in an easy way. Formulas for the definite integral of the
weighted Bernstein polynomi...

We use the matrices of transformations between Chebyshev and Bernstein basis and the matrices of degree elevation and reduction of Chebyshev polynomials to present a simple and efficient method for r times degree elevation and optimal r times degree reduction of Bezier curves with respect to the weighted L2-norm for the interval (0, 1), using the w...

In this paper, we present a method of degree reduction for triangular Bézier surfaces. The approximate and the original triangular Bézier surfaces have common tangent planes at the vertices. We use the least squares method with the L2 and l2 norms to get a closed form for the reduction of the degree and show that both solutions are the same. This s...

We construct Jacobi-weighted orthogonal polynomials
$\mathcal{P}_{n,r}^{(\alpha, \beta ,\gamma)}(u,v,w), \ \alpha,\beta,\gamma > -1, \ \alpha+\beta+\gamma=0$ , on the triangular domain $T$ . We show that these polynomials $\mathcal{P}_{n,r}^{(\alpha, \beta ,\gamma)}(u,v,w)$ over the triangular domain $T$ satisfy the following properties: $\mathcal{...

In this paper, we give the matrix form for the degree elevation of rational triangular Bézier surfaces. We use the L2 measure to find a formula for the distance between two rational triangular Bézier surfaces. A polynomial triangular Bézier surface approximation of a rational triangular Bézier surface based on the least-squares method is given. The...

In this paper we derive the matrix of transformation of the Jacobi
polynomial basis form into the Bernstein polynomial basis of the same degree n and
vice versa. This enables us to combine the superior least-squares performance of the
Jacobi polynomials with the geometrical insight of the Bernstein form. Application to
the inversion of the Bézier c...

Abstract — In paper [4], transformation matrices mapping the Legendre and Bernstein
forms of a polynomial of degree n into each other are derived and examined. In
this paper, we derive a matrix of transformation of Chebyshev polynomials of the first
kind into Bernstein polynomials and vice versa. We also study the stability of these
linear maps and...

A cubic piecewise approximation method is described for planar curves. The order of classical piecewise approximations is improved. The method exploits the freedom in the choice of the parametrization and raises the approximation order to 6. The cubic approximant and the curve have contact of second order. The examples show the simplicity of the co...

In this article, it is shown that the performance of standard quartic Hermite interpolation for planar curves can be improved to achieve a 7th order accuracy. The method is easy to implement. Some examples are given.

We describe an approximation method for planar curves that significantly improves the standard rate obtained by local Taylor approximations. The method exploits the freedom in the choice of the parametrization and achieves the order 4m/3 where m is the degree of the approximating polynomial parametrization. Moreover, we show for a particular set of...

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