中国物理B ›› 2014, Vol. 23 ›› Issue (11): 110207-110207.doi: 10.1088/1674-1056/23/11/110207

• GENERAL • 上一篇    下一篇

A meshless scheme for partial differential equations based on multiquadric trigonometric B-spline quasi-interpolation

高文武a b, 王志刚c   

  1. a School of Economics, Anhui University, Hefei 230411, China;
    b Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China;
    c School of Mathematics and Finance, Fuyang Teachers College, Fuyang 236037, China
  • 收稿日期:2014-04-08 修回日期:2014-05-14 出版日期:2014-11-15 发布日期:2014-11-15
  • 基金资助:

    Project supported by the Shanghai Guidance of Science and Technology, China (Grant No. 12DZ2272800), the Natural Science Foundation of Education Department of Anhui Province, China (Grant No. KJ2013B203), and the Foundation of Introducing Leaders of Science and Technology of Anhui University, China (Grant No. J10117700057).

A meshless scheme for partial differential equations based on multiquadric trigonometric B-spline quasi-interpolation

Gao Wen-Wu (高文武)a b, Wang Zhi-Gang (王志刚)c   

  1. a School of Economics, Anhui University, Hefei 230411, China;
    b Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China;
    c School of Mathematics and Finance, Fuyang Teachers College, Fuyang 236037, China
  • Received:2014-04-08 Revised:2014-05-14 Online:2014-11-15 Published:2014-11-15
  • Contact: Wang Zhi-Gang E-mail:zgw_1122@163.com
  • Supported by:

    Project supported by the Shanghai Guidance of Science and Technology, China (Grant No. 12DZ2272800), the Natural Science Foundation of Education Department of Anhui Province, China (Grant No. KJ2013B203), and the Foundation of Introducing Leaders of Science and Technology of Anhui University, China (Grant No. J10117700057).

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摘要：

Based on the multiquadric trigonometric B-spline quasi-interpolant, this paper proposes a meshless scheme for some partial differential equations whose solutions are periodic with respect to the spatial variable. This scheme takes into account the periodicity of the analytic solution by using derivatives of a periodic quasi-interpolant (multiquadric trigonometric B-spline quasi-interpolant) to approximate the spatial derivatives of the equations. Thus, it overcomes the difficulties of the previous schemes based on quasi-interpolation (requiring some additional boundary conditions and yielding unwanted high-order discontinuous points at the boundaries in the spatial domain). Moreover, the scheme also overcomes the difficulty of the meshless collocation methods (i.e., yielding a notorious ill-conditioned linear system of equations for large collocation points). The numerical examples that are presented at the end of the paper show that the scheme provides excellent approximations to the analytic solutions.

关键词: quasi-interpolation, meshless collocation, periodicity, divided difference

Abstract:

Based on the multiquadric trigonometric B-spline quasi-interpolant, this paper proposes a meshless scheme for some partial differential equations whose solutions are periodic with respect to the spatial variable. This scheme takes into account the periodicity of the analytic solution by using derivatives of a periodic quasi-interpolant (multiquadric trigonometric B-spline quasi-interpolant) to approximate the spatial derivatives of the equations. Thus, it overcomes the difficulties of the previous schemes based on quasi-interpolation (requiring some additional boundary conditions and yielding unwanted high-order discontinuous points at the boundaries in the spatial domain). Moreover, the scheme also overcomes the difficulty of the meshless collocation methods (i.e., yielding a notorious ill-conditioned linear system of equations for large collocation points). The numerical examples that are presented at the end of the paper show that the scheme provides excellent approximations to the analytic solutions.

Key words: quasi-interpolation, meshless collocation, periodicity, divided difference

中图分类号:  (Numerical approximation and analysis)

  • 02.60.-x