数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (1): 387-402.doi: 10.1007/s10473-022-0121-0

• 论文 • 上一篇    下一篇

A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY

郑伟珊1, 陈艳萍2   

  1. 1. College of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China;
    2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China
  • 收稿日期:2020-05-27 修回日期:2021-06-01 出版日期:2022-02-25 发布日期:2022-02-24
  • 通讯作者: Yanping CHEN,E-mail:yanpingchen@scnu.edu.cn E-mail:yanpingchen@scnu.edu.cn
  • 作者简介:Weishan ZHENG,E-mail:weishanzheng@yeah.net
  • 基金资助:
    This work was supported by the State Key Program of National Natural Science Foundation of China (11931003) and the National Natural Science Foundation of China (41974133, 11671157).

A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY

Weishan ZHENG1, Yanping CHEN2   

  1. 1. College of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China;
    2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China
  • Received:2020-05-27 Revised:2021-06-01 Online:2022-02-25 Published:2022-02-24
  • Contact: Yanping CHEN,E-mail:yanpingchen@scnu.edu.cn E-mail:yanpingchen@scnu.edu.cn
  • Supported by:
    This work was supported by the State Key Program of National Natural Science Foundation of China (11931003) and the National Natural Science Foundation of China (41974133, 11671157).

摘要: In this paper, a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay, which contains a weakly singular kernel. We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high order accuracy for the approximation, the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules. In the end, we provide a rigorous error analysis for the proposed method. The spectral rate of convergence for the proposed method is established in both the L-norm and the weighted L2-norm.

关键词: Volterra integro-differential equation, pantograph delay, weakly singular kernel, Jacobi-collocation spectral methods, error analysis, convergence analysis

Abstract: In this paper, a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay, which contains a weakly singular kernel. We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high order accuracy for the approximation, the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules. In the end, we provide a rigorous error analysis for the proposed method. The spectral rate of convergence for the proposed method is established in both the L-norm and the weighted L2-norm.

Key words: Volterra integro-differential equation, pantograph delay, weakly singular kernel, Jacobi-collocation spectral methods, error analysis, convergence analysis

中图分类号: 

  • 65R20