TY - JOUR
T1 - Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions
AU - Wei , Yunxia
AU - Chen , Yanping
JO - Advances in Applied Mathematics and Mechanics
VL - 1
SP - 1
EP - 20
PY - 2012
DA - 2012/04
SN - 4
DO - http://doi.org/10.4208/aamm.10-m1055
UR - https://global-sci.org/intro/article_detail/aamm/103.html
KW - Volterra integro-differential equations, weakly singular kernels, spectral methods,
convergence analysis.
AB - <p style="text-align: justify;">The theory of a class of spectral methods is extended to Volterra 
integro-differential equations which contain a weakly singular 
kernel $(t-s)^{-\mu}$ with $0&lt;\mu&lt;1$. In this work, we consider the
case when the underlying solutions of weakly singular Volterra
integro-differential equations are sufficiently smooth. We provide a
rigorous error analysis for the spectral methods, which shows that
both the errors of approximate solutions and the errors of
approximate derivatives of the solutions decay exponentially in
$L^\infty$-norm and weighted $L^2$-norm. The numerical examples are
given to illustrate the theoretical results.</p>