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Volume 31, Issue 1
A Note on Jacobi Spectral-Collocation Methods for Weakly Singular Volterra Integral Equations with Smooth Solutions

Yanping Chen, Xianjuan Li & Tao Tang

DOI: 10.4208/jcm.1208-m3497

J. Comp. Math., 31 (2013), pp. 47-56.

Published online: 2013-02

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  • Abstract

This work is concerned with spectral Jacobi-collocation methods for Volterra integral equations of the second kind with a weakly singular of the form $(t-s)^{-\alpha}$. When the underlying solutions are sufficiently smooth, the convergence analysis was carried out in [Chen & Tang, J. Comput. Appl. Math., 233 (2009), pp. 938-950]; due to technical reasons the results are restricted to $0<\mu<\frac{1}{2}$. In this work, we will improve the results to the general case $0<\mu<1$ and demonstrate that the numerical errors decay exponentially in the infinity and weighted norms when the smooth solution is involved.

  • Keywords

Volterra integral equations, Convergence analysis, Spectral-collocation methods.

  • AMS Subject Headings

35Q99, 35R35, 65M12, 65M70.

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{JCM-31-47, author = {Yanping Chen, Xianjuan Li and Tao Tang}, title = {A Note on Jacobi Spectral-Collocation Methods for Weakly Singular Volterra Integral Equations with Smooth Solutions}, journal = {Journal of Computational Mathematics}, year = {2013}, volume = {31}, number = {1}, pages = {47--56}, abstract = {

This work is concerned with spectral Jacobi-collocation methods for Volterra integral equations of the second kind with a weakly singular of the form $(t-s)^{-\alpha}$. When the underlying solutions are sufficiently smooth, the convergence analysis was carried out in [Chen & Tang, J. Comput. Appl. Math., 233 (2009), pp. 938-950]; due to technical reasons the results are restricted to $0<\mu<\frac{1}{2}$. In this work, we will improve the results to the general case $0<\mu<1$ and demonstrate that the numerical errors decay exponentially in the infinity and weighted norms when the smooth solution is involved.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1208-m3497}, url = {http://global-sci.org/intro/article_detail/jcm/9720.html} }
TY - JOUR T1 - A Note on Jacobi Spectral-Collocation Methods for Weakly Singular Volterra Integral Equations with Smooth Solutions AU - Yanping Chen, Xianjuan Li & Tao Tang JO - Journal of Computational Mathematics VL - 1 SP - 47 EP - 56 PY - 2013 DA - 2013/02 SN - 31 DO - http://doi.org/10.4208/jcm.1208-m3497 UR - https://global-sci.org/intro/article_detail/jcm/9720.html KW - Volterra integral equations, Convergence analysis, Spectral-collocation methods. AB -

This work is concerned with spectral Jacobi-collocation methods for Volterra integral equations of the second kind with a weakly singular of the form $(t-s)^{-\alpha}$. When the underlying solutions are sufficiently smooth, the convergence analysis was carried out in [Chen & Tang, J. Comput. Appl. Math., 233 (2009), pp. 938-950]; due to technical reasons the results are restricted to $0<\mu<\frac{1}{2}$. In this work, we will improve the results to the general case $0<\mu<1$ and demonstrate that the numerical errors decay exponentially in the infinity and weighted norms when the smooth solution is involved.

Yanping Chen, Xianjuan Li and Tao Tang. (2013). A Note on Jacobi Spectral-Collocation Methods for Weakly Singular Volterra Integral Equations with Smooth Solutions. Journal of Computational Mathematics. 31 (1). 47-56. doi:10.4208/jcm.1208-m3497
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