TY - JOUR
T1 - A Note on Jacobi Spectral-Collocation Methods for Weakly Singular Volterra Integral Equations with Smooth Solutions
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 - <p style="text-align: justify;">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 &amp; Tang, J. Comput. Appl. Math., 233 (2009), pp. 938-950]; due to technical reasons the results are restricted to $0&lt;\mu&lt;\frac{1}{2}$. In this work, we will improve the results to the general case $0&lt;\mu&lt;1$ and demonstrate that the numerical errors decay exponentially in the infinity and weighted norms when the smooth solution is involved.</p>