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

Acta Mathematica Scientia volume 42, pages 387–402 (2022)Cite this article

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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References

  1. Ali I, Brunner H, Tang T, Spectral methods for pantograph-type differential and integral equations with multiple delays. Front Math China, 2009, 4: 49–61

    MathSciNet  Article  Google Scholar 

  2. Canuto C, Hussaini M Y, Quarteroni A, et al. Spectral Methods Fundamentals in Single Domains. Springer-Verlag, 2006

  3. Chen Y P, Tang T, Spectral methods for weakly singular Volterra integral equations with smooth solutions. J Comput Appl Math, 2009, 233: 938–950

    MathSciNet  Article  Google Scholar 

  4. Chen Y P, Tang T, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math Comp, 2010, 79: 147–167

    MathSciNet  Article  Google Scholar 

  5. Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences 93. 2nd ed. Heidelberg: Springer-Verlag, 1998

    Book  Google Scholar 

  6. Enright W H, Hu M, Continuous Runge-Kutta methods for neutral Volterra integro-differential equations with delay. Appl Numer Math, 1997, 24: 175–190

    MathSciNet  Article  Google Scholar 

  7. Gan S Q, Dissipativity of θ-methods for nonlinear Volterra delay-integro-differential equations. J Comput Appl Math, 2007, 206: 898–907

    MathSciNet  Article  Google Scholar 

  8. Guo B Y, Wang L L, Jacobi interpolation approximations and their applications to singular differential equations. Adv Comput Math, 2001, 14: 227–276

    MathSciNet  Article  Google Scholar 

  9. Guo B Y, Shen J, Wang L L, Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J Sci Comput, 2006, 27: 305–322

    MathSciNet  Article  Google Scholar 

  10. Ishiwata E, Muroya Y, On collocation methods for delay differential and Volterra integral equations with proportional delay. Front Math China, 2009, 4(1): 89–111

    MathSciNet  Article  Google Scholar 

  11. Kufner A, Persson L E. Weighted Inequalities of Hardy Type. New York: World Scientific, 2003

    Book  Google Scholar 

  12. Mastroianni G, Occorsio D. Optimal systems of nodes for Lagrange interpolation on bounded intervals, A survey. J Comput Appl Math, 2001, 134: 325–341

    MathSciNet  Article  Google Scholar 

  13. Nevai P. Mean convergence of Lagrange interpolation, III. Trans Amer Math Soc, 1984, 282: 669–698

    MathSciNet  Article  Google Scholar 

  14. Ragozin D L, Polynomial approximation on compact manifolds and homogeneous spaces. Trans Amer Math Soc, 1970, 150: 41–53

    MathSciNet  Article  Google Scholar 

  15. Ragozin D L, Constructive polynomial approximation on spheres and projective spaces. Trans Amer Math Soc, 1971, 162: 157–170

    MathSciNet  MATH  Google Scholar 

  16. Shen J, Tang T. Spectral and High-Order Methods with Applications. Beijing: Science Press, 2006

    MATH  Google Scholar 

  17. Wang W S, Li S F, Convergence of Runge-Kutta methods for neutral Volterra delay-integro-differential equations. Front Math China, 2009, 4: 195–216

    MathSciNet  Article  Google Scholar 

  18. Wei Y X, Chen Y P, Legendre spectral collocation methods for pantograph Volterra delay integro-differential equations. J Sci Comput, 2012, 53: 672–688

    MathSciNet  Article  Google Scholar 

  19. Wei Y X, Chen Y P, Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions. Adv Appl Math Mech, 2012, 4: 1–20

    MathSciNet  Article  Google Scholar 

  20. Wei Y X, Chen Y P, Legendre spectral collocation method for Volterra-Hammerstein integral equation of the second kind. Acta Math Sci, 2017, 37B(4): 1105–1114

    MathSciNet  Article  Google Scholar 

  21. Wu S F, Gan S Q, Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations. Comput Math Appl, 2008, 55: 2426–2443

    MathSciNet  Article  Google Scholar 

  22. Yang Y, Chen Y P, Huang Y Q, Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations. Acta Math Sci, 2014, 34B(3): 673–690

    MathSciNet  Article  Google Scholar 

  23. Zhang C J, Vandewalle S, Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. IMA J Numer Anal, 2004, 24: 193–214

    MathSciNet  Article  Google Scholar 

  24. Zhao J J, Xu Y, Liu M Z, Stability analysis of numerical methods for linear neutral Volterra delay-integro-differential system. Appl Math Comput, 2005, 167: 1062–1079

    MathSciNet  MATH  Google Scholar 

  25. Zheng W S, Chen Y P, Numerical analysis for Volterra integral equation with two kinds of delay. Acta Math Sci, 2019, 39B(2): 607–617

    MathSciNet  Article  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China

    Weishan Zheng

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Yanping Chen

Authors
  1. Weishan Zheng
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  2. Yanping Chen
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Corresponding authors

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