Ahmad Abdul Majid

The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems

In this paper, a collocation finite difference scheme based on new cubic trigonometric B-spline is developed and analyzed for the numerical solution of a one-dimensional hyperbolic equation (wave equation) with non-local conservation condition. The usual finite difference scheme is used to discretize the time derivative while a cubic trigonometric B-spline is utilized as an interpolation function in the space dimension. The scheme is shown to be unconditionally stable using the von Neumann (Fourier) method. The accuracy of the proposed scheme is tested by using it for several test problems. The numerical results are found to be in good agreement with known exact solutions and with existing schemes in literature.

Monotonicity-preserving C2 rational cubic spline for monotone data

Abstract Designers in industries need to generate splines which can interpolate the data points in such a way that they preserve the inherited shape characteristics (positivity, monotonicity, convexity) of data. Among the properties that the spline for curves and surfaces need to satisfy, smoothness and shape preservation of given data are mostly needed by all the designers. In this paper, a rational cubic function with three shape parameters has been developed. Data dependent sufficient constraints are derived for one of these shape parameters to preserve the inherited shape feature like monotonicity of data. Remaining two shape parameters are left free for designer to refine the shape of the monotone curve as desired. Numerical examples and interpolation error analysis show that the interpolant is not only C 2 , local, computationally economical and visually pleasant but also smooth. The error of rational cubic function is also calculated when the arbitrary function being interpolated is C 3 in an interpolating interval. The order of approximation of interpolant is O ( h i 3 ) .

Numerical method using cubic B-spline for the heat and wave equation

In the present paper, the cubic B-splines method is considered for solving one-dimensional heat and wave equations. A typical finite difference approach had been used to discretize the time derivative while the cubic B-spline is applied as an interpolation function in the space dimension. The accuracy of the method for both equations is discussed. The efficiency of the method is illustrated by some test problems. The numerical results are found to be in good agreement with the exact solution.

An Automatic Generation of G^1 Curve Fitting of Arabic Characters

An Arabic font is difficult to fit as it is cursive in character, having varying curves and cusps. Here, the Arabic character is represented as an outline font fitted with G1 rational Bezier cubic curves. As a method in reverse engineering, the Arabic character is created by way of digitizing an image that already exists and then fitting G1 curves automatically to the outline of the digitized image. The outline font representation is done in several phases - contour extraction of font image, corner points detection and lastly contour segment fitting. Image is considered as binary and boundary is obtained accordingly. Eigenvalues of covariance matrix and the concept of region of support are employed to search for the corners of the Arabic characters which are of varying degrees of smoothness. G 1 rational Bezier cubics, iteratively determined, are used in the last step. The weights are adjusted automatically to get curves that are as close as need be to the digitized data points. This technique can be extended to visualizing outlines of other contour-based images automatically

Fuzzy Geometric Modeling

Fuzzy geometric modeling provides a useful tool to introduce uncertainty into mathematical spline model. In this paper a new concept of a geometric modeling is presented, based on the theory of fuzzy numbers. By the notion of fuzzy number we introduce fuzzy control point for fuzzy curve and fuzzy surface model for CAGD. We study the properties concerning approximation of fuzzy control points by means of fuzzy Bezier, fuzzy B-spline and fuzzy NURBS.

Positivity-preserving rational bi-cubic spline interpolation for 3D positive data

This paper deals with the shape preserving interpolation problem for visualization of 3D positive data. A required display of 3D data looks smooth and pleasant. A rational bi-cubic function involving six shape parameters is presented for this objective which is an extension of piecewise rational function in the form of cubic/quadratic involving three shape parameters. Simple data dependent constraints for shape parameters are derived to conserve the inherited shape feature (positivity) of 3D data. Remaining shape parameters are left free for designer to modify the shape of positive surface as per industrial needs. The interpolant is not only local, C 1 but also it is a computationally economical in comparison with existing schemes. Several numerical examples are supplied to support the worth of proposed interpolant.

Cubic B-Spline Collocation Method for One-Dimensional Heat and Advection-Diffusion Equations

Numerical solutions of one-dimensional heat and advection-diffusion equations are obtained by collocation method based on cubic B-spline. Usual finite difference scheme is used for time and space integrations. Cubic B-spline is applied as interpolation function. The stability analysis of the scheme is examined by the Von Neumann approach. The efficiency of the method is illustrated by some test problems. The numerical results are found to be in good agreement with the exact solution.

A new homotopy function for solving nonlinear equations

We develop a new homotopy function, H*(x,t) to be used in conjunction with the homotopy continuation method. We test the new function on several examples in literature survey. The results obtained indicate the superior accuracy of the new homotopy function compared with the standard homotopy function. This new homotopy function is based on the De Casteljau Algorithms which is used in Computer Aided Geometric Design.

Numerical solution of the coupled viscous Burgers equations by Chebyshev-Legendre Pseudo-Spectral method

In this paper, we consider Chebyshev-Legendre Pseudo-Spectral (CLPS) method for solving coupled viscous Burgers (VB) equations. A leapfrog scheme is used in time direction, while CLPS method is used for space direction. Chebyshev-Gauss-Lobatto (CGL) nodes are used for practical computation. The error estimates of semi-discrete and fully-discrete of CLPS method for coupled VB equations are obtained by energy estimation method. The numerical results of the present method are compared with the exact solution for two test problems.

Fuzzy set in geometric modeling

Geometric modeling involving uncertainty of data is the major problem in CAGD. By using the theory of fuzzy set and its properties, the issues of uncertainty can be solved through the concept of fuzzy numbers. We apply fuzzy numbers as uncertainty data and defined a new kind of fuzzy control points. With this type of points we then introduced a fuzzy Bezier curve and fuzzy B-spline. By these definitions, we construct a curve of such fuzzy spline and given an example of fuzzy Bezier curve and fuzzy B-spline in the context of CAGD at the end of this concept papers.