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Volterra Integral Equations with Vanishing Delay

Received: 29 March 2015     Accepted: 6 May 2015     Published: 27 May 2015
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Abstract

In this article, we use a Chebyshev spectral-collocation method to solve the Volterra integral equations with vanishing delay. Then a rigorous error analysis provided by the proposed method shows that the numerical error decay exponentially in the infinity norm and in the Chebyshev weighted Hilbert space norm. Numerical results are presented, which confirm the theoretical predicition of the exponential rate of convergence.

Published in Applied and Computational Mathematics (Volume 4, Issue 3)
DOI 10.11648/j.acm.20150403.18
Page(s) 152-161
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Chebyshev Spectral-Collocation Method, Volterra Integral Equations, Vanishing Delay, Error Estimate, Convergence Analysis

References
[1] T. Tang, X. Xu, J. Cheng, On spectral methods for Volterra type integral equations and the convergence analysis, J. Comput. Math. 26 (2008) 825-837.
[2] Y. Chen, and T. Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equation with a weakly singular kernel, Math. Comput. 79(2010), pp. 147–167.
[3] Z. Wan, Y. Chen, and Y. Huang, Legendre spectral Galerkin method for second-kind Volterraintegral equations, Front. Math. China, 4(2009), pp. 181–193.
[4] Z. Xie, X. Li and T. Tang, Convergence Analysis of Spectral Galerkin Methods for Volterra Type Integral Equations, J. Sci. Comput, 2012.
[5] Z. Gu and Y. Chen, Chebyshev spectral collocation method for Volterra integral equations [D•Master's Thesis], Contemporary Mathematics, Volume 586, 2013, pp. 163-170.
[6] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral method fundamentalsin single domains, Spring-Verlag, 2006.
[7] J. Shen, and T. Tang, Spectral and high-order methods with applications, Science Press, Beijing, 2006.
[8] D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, 1989.
[9] A. Kufner, and L. E. Persson, Weighted inequality of Hardy’s Type, World scientific, NewYork, 2003.
[10] P. Nevai, Mean convergence of Lagrange interpolation, III, Trans. Amer. Math. Soc., 282(1984), pp. 669–698.
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  • APA Style

    Xiaoxuan Li, Weishan Zheng, Jiena Wu. (2015). Volterra Integral Equations with Vanishing Delay. Applied and Computational Mathematics, 4(3), 152-161. https://doi.org/10.11648/j.acm.20150403.18

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    ACS Style

    Xiaoxuan Li; Weishan Zheng; Jiena Wu. Volterra Integral Equations with Vanishing Delay. Appl. Comput. Math. 2015, 4(3), 152-161. doi: 10.11648/j.acm.20150403.18

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    AMA Style

    Xiaoxuan Li, Weishan Zheng, Jiena Wu. Volterra Integral Equations with Vanishing Delay. Appl Comput Math. 2015;4(3):152-161. doi: 10.11648/j.acm.20150403.18

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  • @article{10.11648/j.acm.20150403.18,
      author = {Xiaoxuan Li and Weishan Zheng and Jiena Wu},
      title = {Volterra Integral Equations with Vanishing Delay},
      journal = {Applied and Computational Mathematics},
      volume = {4},
      number = {3},
      pages = {152-161},
      doi = {10.11648/j.acm.20150403.18},
      url = {https://doi.org/10.11648/j.acm.20150403.18},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150403.18},
      abstract = {In this article, we use a Chebyshev spectral-collocation method to solve the Volterra integral equations with vanishing delay. Then a rigorous error analysis provided by the proposed method shows that the numerical error decay exponentially in the infinity norm and in the Chebyshev weighted Hilbert space norm. Numerical results are presented, which confirm the theoretical predicition of the exponential rate of convergence.},
     year = {2015}
    }
    

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    T1  - Volterra Integral Equations with Vanishing Delay
    AU  - Xiaoxuan Li
    AU  - Weishan Zheng
    AU  - Jiena Wu
    Y1  - 2015/05/27
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    N1  - https://doi.org/10.11648/j.acm.20150403.18
    DO  - 10.11648/j.acm.20150403.18
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
    SP  - 152
    EP  - 161
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20150403.18
    AB  - In this article, we use a Chebyshev spectral-collocation method to solve the Volterra integral equations with vanishing delay. Then a rigorous error analysis provided by the proposed method shows that the numerical error decay exponentially in the infinity norm and in the Chebyshev weighted Hilbert space norm. Numerical results are presented, which confirm the theoretical predicition of the exponential rate of convergence.
    VL  - 4
    IS  - 3
    ER  - 

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Author Information
  • Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, China

  • Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, China

  • Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, China

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