Ahmad Abd. Majid

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.

A New Trigonometric Spline Approach to Numerical Solution of Generalized Nonlinear Klien-Gordon Equation

The generalized nonlinear Klien-Gordon equation plays an important role in quantum mechanics. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline is presented for the approximate solution of this equation with Dirichlet boundary conditions. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Several examples are discussed to exhibit the feasibility and capability of the approach. The absolute errors and error norms are also computed at different times to assess the performance of the proposed approach and the results were found to be in good agreement with known solutions and with existing schemes in literature.

Local convexity-preserving C2 rational cubic spline for convex data.

We present the smooth and visually pleasant display of 2D data when it is convex, which is contribution towards the improvements over existing methods. This improvement can be used to get the more accurate results. An attempt has been made in order to develop the local convexity-preserving interpolant for convex data using C 2 rational cubic spline. It involves three families of shape parameters in its representation. Data dependent sufficient constraints are imposed on single shape parameter to conserve the inherited shape feature of data. Remaining two of these shape parameters are used for the modification of convex curve to get a visually pleasing curve according to industrial demand. The scheme is tested through several numerical examples, showing that the scheme is local, computationally economical, and visually pleasing.

Numerical Method Using Cubic Trigonometric B-Spline Technique for Nonclassical Diffusion Problems

A new two-time level implicit technique based on cubic trigonometric B-spline is proposed for the approximate solution of a nonclassical diffusion problem with nonlocal boundary constraints. The standard finite difference approach is applied to discretize the time derivative while cubic trigonometric B-spline is utilized as an interpolating function in the space dimension. The technique is shown to be unconditionally stable using the von Neumann method. Several numerical examples are discussed to exhibit the feasibility and capability of the technique. The and error norms are also computed at different times for different space size steps to assess the performance of the proposed technique. The technique requires smaller computational time than several other methods and the numerical results are found to be in good agreement with known solutions and with existing schemes in the literature.

Local Convexity Shape-Preserving Data Visualization by Spline Function

The main purpose of this paper is the visualization of convex data that results in a smooth, pleasant, and interactive convexity-preserving curve. The rational cubic function with three free parameters is constructed to preserve the shape of convex data. The free parameters are arranged in a way that two of them are left free for user choice to refine the convex curve as desired, and the remaining one free parameter is constrained to preserve the convexity everywhere. Simple data-dependent constraints are derived on one free parameter, which guarantee to preserve the convexity of curve. Moreover, the scheme under discussion is, 𝐶1 flexible, simple, local, and economical as compared to existing schemes. The error bound for the rational cubic function is 𝑂(ℎ3).

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.

Conic Curve Fitting Using Particle Swarm Optimization: Parameter Tuning

We solve curve fitting problems using Particle Swarm Optimization (PSO). PSO is used to optimize control points and weights of two conic curves to a set of data points. PSO is used to find the best middle control point and weight for both conic curves to provide piecewise conics that preserve G^1 continuity. We present numerical result using parameter changes in PSO scheme. We obtain appropriate parameter values of PSO that provide best error and fastest time to solve curve fitting problem.

The Representation of Circular Arc by Using Rational Cubic Timmer Curve

In CAD/CAM systems, rational polynomials, in particular the Bezier or NURBS forms, are useful to approximate the circular arcs. In this paper, a new representation method by means of rational cubic Timmer (RCT) curves is proposed to effectively represent a circular arc. The turning angle of a rational cubic Bezier and rational cubic Ball circular arcs without negative weight is still not more than and , respectively. The turning angle of proposed approach is more than Bezier and Ball circular arcs with easier calculation and determination of control points. The proposed method also provides the easier modification in the shape of circular arc showing in several numerical examples.

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.

Solving Boussinesq equation using quintic B-spline and quintic trigonometric B-spline interpolation methods

The quintic B-spline (QBS) and quintic trigonometric B-spline (QTBS) functions are used to set up the collocation methods in finding solutions for the Boussinesq equation. The QBS and QTBS are applied as interpolating functions in the spatial dimension while the finite difference method (FDM) is used to discretize the time derivative. The nonlinear Boussinesq equation is linearized using Taylor’s expansion. The von Neumann stability analysis is used to analyze the schemes and they are shown to be conditionally stable. In order to demonstrate the capability of the schemes, some problems are solved and compared with the analytical solutions and generated results from the FDM. The proposed numerical schemes are found to be accurate.