Volterra Integral Equations with Vanishing Delay

Xiaoxuan Li; Weishan Zheng; Jiena Wu

doi:doi:10.11648/j.acm.20150403.18

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Volterra Integral Equations with Vanishing Delay

Xiaoxuan Li, Weishan Zheng, Jiena Wu

Published in Applied and Computational Mathematics (Volume 4, Issue 3)

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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TY  - JOUR
T1  - Volterra Integral Equations with Vanishing Delay
AU  - Xiaoxuan Li
AU  - Weishan Zheng
AU  - Jiena Wu
Y1  - 2015/05/27
PY  - 2015
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

Xiaoxuan Li

Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, China
Weishan Zheng

Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, China
Jiena Wu

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

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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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Volterra Integral Equations with Vanishing Delay
AU  - Xiaoxuan Li
AU  - Weishan Zheng
AU  - Jiena Wu
Y1  - 2015/05/27
PY  - 2015
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  -

Copy | Download