TY - JOUR
T1 - A Robust Hybrid Spectral Method for Nonlocal Problems with Weakly Singular Kernels
AU - Zhang , Chao
AU - Yao , Guoqing
AU - Chen , Sheng
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 1041
EP - 1062
PY - 2022
DA - 2022/10
SN - 15
DO - http://doi.org/10.4208/nmtma.OA-2022-0006s 
UR - https://global-sci.org/intro/article_detail/nmtma/21089.html
KW - Nonlocal problem, Volterra integral, spectral element method, log orthogonal function, Legendre polynomial, weak singularity, exponential convergence.
AB - <p style="text-align: justify;">In this paper, we propose a hybrid spectral method for a type of nonlocal
problems, nonlinear Volterra integral equations (VIEs) of the second kind. The main
idea is to use the shifted generalized Log orthogonal functions (GLOFs) as the basis
for the first interval and employ the classical shifted Legendre polynomials for other
subintervals. This method is robust for VIEs with weakly singular kernel due to the
GLOFs can efficiently approximate one-point singular functions as well as smooth
functions. The well-posedness and the related error estimates will be provided.
Abundant numerical experiments will verify the theoretical results and show the
high-efficiency of the new hybrid spectral method.</p>