Geometric Properties and Algorithms for Rational q-Bézier Curves and Surfaces

In this paper, properties and algorithms of q-Bézier curves and surfaces are analyzed. It is proven that the only q-Bézier and rational q-Bézier curves satisfying the boundary tangent property are the Bézier and rational Bézier curves, respectively. Evaluation algorithms formed by steps in barycentric form for rational q-Bézier curves and surfaces are provided.

Rational Spectral Collocation Combined with the Singularity Separated Method for a System of Singularly Perturbed Boundary Value Problems

A novel rational spectral collocation method is presented combined with the singularity-separated technique for a system of singularly perturbed boundary value problems. The solution is expressed as u=w+v, where w is the solution of the corresponding auxiliary boundary value problem and v is a singular correction with explicit expressions. The rational spectral collocation method in barycentric form with the sinh transformation is applied to solve the auxiliary third boundary problem. The parameters of the singular correction can be determined by the boundary conditions of the original problem. Numerical experiments are carried out to support theoretical results and provide a favorable comparison with research results of other work.

Lebesgue Constant Using Sinc Points

Lebesgue constant for Lagrange approximation at Sinc points will be examined. We introduce a new barycentric form for Lagrange approximation at Sinc points. Using Thiele’s algorithm we show that the Lebesgue constant grows logarithmically as the number of interpolation Sinc points increases. A comparison between the obtained upper bound of Lebesgue constant using Sinc points and other upper bounds for different set of points, like equidistant and Chebyshev points, is introduced.

Fourier Cosine Differential Quadrature Method for Beam and Plate Problems

In this paper, we combined the Fourier cosine series and differential quadrature method (DQM) in barycentric form to develop a new method (FCDQM), which is applied to the 1D fourth order beam problem and the 2D thin isotropic plate problems. Furthermore, we solved the complex boundary conditions on irregular domains with DQM directly. The numerical results illustrate the stability, validity and good accuracy of the method in treating this class of engineering problems.