Muhammad Abbas

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 a Strongly Coupled Reaction-Diffusion System

In this paper, a numerical method for the solution of a strongly coupled reaction-diffusion system, with suitable initial and Neumann boundary conditions, by using cubic B-spline collocation scheme on a uniform grid is presented. The scheme is based on the usual finite difference scheme to discretize the time derivative while cubic B-spline is used as an interpolation function in the space dimension. The scheme is shown to be unconditionally stable using the von Neumann method. The accuracy of the proposed scheme is demonstrated by applying it on a test problem. The performance of this scheme is shown by computing and error norms for different time levels. The numerical results are found to be in good agreement with known exact solutions.

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.

The G2 and C2 rational quadratic trigonometric Bézier curve with two shape parameters with applications

The rational quadratic trigonometric Bezier curve with two shape parameters is presented in this paper, which is new in literature. The purposed curve inherits all the geometric properties of the traditional rational quadratic Bezier curve. The presence of shape parameters provides a control on the shape of the curve more than that of traditional Bezier curve. Moreover the weight offers an additional control on the curve. Simple constraints for shape parameters are derived using the end points curvature so that their values always fall within the defined range. The composition of two segments of curve using G^2 and C^2 continuity is given. The new curves can accurately represent some conics and best approximates the traditional rational quadratic Bezier curve.

Spur Gear Tooth Design and Transition Curve as a Spiral Using Cubic Trigonometric Bezier Function

In this paper, we construct an S-shaped transition curve and spur gear tooth model. The attempt has been made to use the circle to circle technique an S-Shaped transition curve by using Cubic Trigonometric Bezier function (TBezier, for short) with two shape parameters . The shape parameters are used to control the S-shaped transition curves and provide the more flexibility for the interactive design. We have used two curve segments for the completion of single gear teeth instead of four curve segments. This model is the introduction of new cost effective, more economical and computationally easy a more efficient for CAD and CAE systems than ever before. The smoothness of the curve is G2 continuity.

Application of Hybrid Cubic B-Spline Collocation Approach for Solving a Generalized Nonlinear Klien-Gordon Equation

The generalized nonlinear Klien-Gordon equation is important in quantum mechanics and related fields. In this paper, a semi-implicit approach based on hybrid cubic B-spline is presented for the approximate solution of the nonlinear Klien-Gordon equation. The usual finite difference approach is used to discretize the time derivative while hybrid cubic B-spline is applied as an interpolating function in the space dimension. The results of applications to several test problems indicate good agreement with known solutions.

A Rational Spline for Preserving the Shape of Positive Data

—In Computer Aided Geometric Design (CAGD), it is often needed to produce a positivity–preserving curve according to the given positive data. The main focus of this work is to address the problem of visualizing positive data in such a way that its display looks smooth and pleasant. A rational cubic spline function with three shape parameters has been developed. Simple data dependent constraints are derived for single shape parameter to preserve the positivity through positive data. Remaining two shape parameters are provided extra freedom to user for modification of curves as desired. The scheme is local, computationally economical and time saving as compared to existing schemes. The curve scheme under discussion is attained 1 C continuity.

The Rational Quadratic Trigonometric Bézier Curve with Two Shape Parameters

In this paper, a newly constructed rational quadratic trigonometric Bézier curve with two shape parameters is presented. The purposed curve enjoys all the geometric properties of the traditional rational quadratic Bézier curve. The local control on the shape of the curve can be attained by altering the values of the shape parameters as well as the weight. The curve exactly represents some quadratic trigonometric curves such as the arc of an ellipse and the arc of a circle and best approximates the ordinary rational quadratic Bézier curve.

Shape preserving rational cubic Ball interpolation for positive data

In this work, a C1 piecewise rational cubic Ball curve function with three shape parameters is presented. The shape parameters are utilized to control and modify the shape of data as required. The function is used to visualize the positive data and conserves the positivity of data. Numerical examples show that the present interpolant is quite efficient.