TY - JOUR
T1 - A Wavelet Method for the Fredholm Integro-Differential Equations with Convolution Kernel
AU -  , Xiao-Qing Jin
AU - Sin , Vai-Kuong
AU - Yuan , Jin-Yun
JO - Journal of Computational Mathematics
VL - 4
SP - 435
EP - 440
PY - 1999
DA - 1999/08
SN - 17
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/9114.html
KW - Fredholm integro-differential equation, Kernel, Wavelet transform,
Toeplitz matrix, Hankel matrix, Sobolev space, PCG method.
AB - <p style="text-align: justify;">We study the Fredholm integro-differential equation $$D^{2s}_xσ(x) +
\int_{-∞}^{+∞}&nbsp;&nbsp;k(x &nbsp;y)σ(y)dy = g(x)$$ by the wavelet method. Here $σ(x)$ is the unknown fun
tion to be found, $k(y)$ is
a convolution kernel and $g(x)$ is a given function. Following the idea in [7], the
equation is discretized with respect to two different wavelet bases. We then have
two different linear systems. One of them is a Toeplitz-Hankel system of the form $(H_n + T_n)x = b$ where $T_n$ is a Toeplitz matrix and $H_n$ is a Hankel matrix. The
other one is a system $(B_n + C_n)y = d$ with condition number $k = O(1)$ after a
diagonal scaling. By using the preconditioned conjugate gradient (PCG) method
with the fast wavelet transform (FWT) and the fast iterative Toeplitz solver, we can solve the systems in $O(n$ ${\rm log}$ $n)$ operations.</p>