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A spectral method for a weakly singular Volterra integro-differential equation with pantograph delay

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.

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Correspondence to Weishan Zheng or Yanping Chen.

Additional information

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).

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Zheng, W., Chen, Y. A spectral method for a weakly singular Volterra integro-differential equation with pantograph delay. Acta Math Sci 42, 387–402 (2022). https://doi.org/10.1007/s10473-022-0121-0

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  • DOI: https://doi.org/10.1007/s10473-022-0121-0

Key words

  • Volterra integro-differential equation
  • pantograph delay
  • weakly singular kernel
  • Jacobi-collocation spectral methods
  • error analysis
  • convergence analysis

2010 MR Subject Classification

  • 65R20
  • 45E05