@Article{JCM-17-4,
author = {Xiao-Qing, Jin and Vai-Kuong, Sin and Yuan, Jin-Yun},
title = {A Wavelet Method for the Fredholm Integro-Differential Equations with Convolution Kernel},
journal = {Journal of Computational Mathematics},
year = {1999},
volume = {17},
number = {4},
pages = {435--440},
abstract = {<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>},
issn = {1991-7139},
doi = {https://doi.org/1999-JCM-9114},
url = {https://global-sci.com/article/85639/a-wavelet-method-for-the-fredholm-integro-differential-equations-with-convolution-kernel}
}