Ren-Hong Wang

A novel numerical scheme for solving Burgers’ equation

Abstract In this paper, we propose a novel numerical scheme for solving Burgers’ equation. The scheme is based on a cubic spline quasi-interpolant and multi-node higher order expansion, which make the algorithm simple and easy to implement. The numerical experiments show that the proposed method produces high accurate results.

Numerical solution of one-dimensional Sine–Gordon equation using high accuracy multiquadric quasi-interpolation ☆

Abstract In this paper, we propose a numerical scheme to solve one-dimensional Sine–Gordon equation related to many scientific research topics by using high accuracy multiquadric quasi-interpolation. We use the derivatives of a multiquadric quasi-interpolant to approximate the spatial derivatives, and a finite difference to approximate the temporal derivative. The advantages of the scheme are that it is meshfree, and in each time step only a multiquadric quasi-interpolant is employed, so that the algorithm is very easy to implement. The accuracy of our scheme is demonstrated by some test problems.

An approach for designing a developable surface through a given line of curvature

Developable surface and line of curvature play an important role in geometric design and surface analysis. This paper proposes a new method to construct a developable surface possessing a given curve as the line of curvature of it. We analyze the necessary and sufficient conditions when the resulting developable surface is a cylinder, cone or tangent surface. Finally, we illustrate the convenience and efficiency of this method by some representative examples.

Design and G1 connection of developable surfaces through Bézier geodesics

Abstract For potential application in shoemaking and garment manufacture industries, the G 1 connection of (1,xa0 k ) developable surfaces with abutting geodesic is important. In this paper, we discuss the developable surface which contains a given 3D Bezier curve as geodesic and prove the corresponding conclusions in detail. Primarily we study G 1 connection of developable surfaces through abutting cubic Bezier geodesics and give some examples.

Analysis of a 6-point binary subdivision scheme

Abstract In this paper, a linear 6-point binary approximating subdivision scheme with support [−6,xa05] is fully investigated. It is shown that the scheme is simple and has elegant properties. We prove that the scheme can have high order continuity, polynomial reproduction and convexity preserving properties simultaneously. Furthermore, we explore the trade-offs among these properties. Examples are given to demonstrate the efficiency and flexibility of the scheme.

Bivariate polynomial and continued fraction interpolation over ortho-triples

Abstract By means of the barycentric coordinates expression of the interpolating polynomial over each ortho-triple, some properties are obtained. Moreover, the explicit coefficients in terms of B-net for one ortho-triple, and two ortho-triples are worked out, respectively. Thus the computation of multiple integrals can be converted into the sum of the coefficients in terms of the B-net over triangular domain much effectively and conveniently. Based on a new symmetrical algorithm of partial inverse differences, a novel continued fractions interpolation scheme is presented over arbitrary ortho-triples in R2, which is a bivariate osculatory interpolation formula with one-order partial derivatives at all corner points in the ortho-triples. Furthermore, its characterization theorem is presented by three-term recurrence relations. The new scheme is advantageous over the polynomial one with some numerical examples.

Algebraic approximants to exp(z) and applications in construction of difference schemes of first order ODE

In this paper, an approximation to exponential function exp(z) by algebraic functions is given. Its applications in construction of difference schemes of first order ODE are also shown.

Real root classification of parametric spline functions

Abstract The real root classification of a given parametric spline function is a collection of possible cases of its real root distribution on every interval, together with the conditions of its coefficients must be satisfied for each case. This paper presents an algorithm to deal with the real root classification of a given parametric spline function. Two examples are provided to illustrate the proposed algorithm is flexible.

Numerical integration based on a multilevel quartic quasi-interpolation operator

Quasi-interpolation is very important in the study of the scattered data approximation, numerical integration and numerical solutions of differential equations. In this paper, we proposed a multilevel quasi-interpolation scheme by using B-spline basis functions on quartic spline space. Moreover, the proposed scheme is applied to the numerical integration of two-dimensional singular integrals. Some numerical results with contrast to the existing methods are provided.

A generalization of surface family with common line of curvature

We can represent the surface with a linear combination of the components of Frenet-Serret frame. Based on this representation, in the work of Li et al. [C.-Y. Li, R.-H. Wang, C.-G. Zhu, Parametric representation of a surface pencil with a common line of curvature, Comput. Aided Des. 43 (9) (2011) 1110-1117], we derive the necessary and sufficient condition on the marching-scale functions for which the given curve is a line of curvature of the resulting surface. For convenience, we assumed the marching-scale functions can be decomposed into two factors. In this paper, we derive the sufficient condition for the given curve as a line of curvature of the surface when the marching-scale functions are in more general expressions. Finally, we give some representative examples to illustrate the convenience and efficiency of this method.