TY - JOUR
T1 - Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel
AU - Liu , Xiong
AU - Chen , Yanping
JO - Advances in Applied Mathematics and Mechanics
VL - 6
SP - 1506
EP - 1524
PY - 2017
DA - 2017/09
SN - 9
DO - http://doi.org/10.4208/aamm.OA-2016-0049
UR - https://global-sci.org/intro/article_detail/aamm/10190.html
KW - Chebyshev collocation method, Volterra integral equations, spectral rate of convergence, Hölder continuity.
AB - <p style="text-align: justify;">In this paper, a Chebyshev-collocation spectral method is developed for
Volterra integral equations (VIEs) of second kind with weakly singular kernel. We first
change the equation into an equivalent VIE so that the solution of the new equation
possesses better regularity. The integral term in the resulting VIE is approximated by
Gauss quadrature formulas using the Chebyshev collocation points. The convergence
analysis of this method is based on the Lebesgue constant for the Lagrange interpolation
polynomials, approximation theory for orthogonal polynomials, and the operator
theory. The spectral rate of convergence for the proposed method is established in the
L<sup>∞</sup>-norm and weighted L<sup>2</sup>-norm. Numerical results are presented to demonstrate the
effectiveness of the proposed method.</p>