arrow
Volume 17, Issue 4
A Wavelet Method for the Fredholm Integro-Differential Equations with Convolution Kernel

Xiao-Qing Jin , Vai-Kuong Sin & Jin-Yun Yuan

J. Comp. Math., 17 (1999), pp. 435-440.

Published online: 1999-08

Export citation
  • Abstract

We study the Fredholm integro-differential equation $$D^{2s}_xσ(x) + \int_{-∞}^{+∞}  k(x  y)σ(y)dy = g(x)$$ 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.

  • AMS Subject Headings

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{JCM-17-435, author = { , Xiao-Qing JinSin , Vai-Kuong 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 = {

We study the Fredholm integro-differential equation $$D^{2s}_xσ(x) + \int_{-∞}^{+∞}  k(x  y)σ(y)dy = g(x)$$ 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.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9114.html} }
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 -

We study the Fredholm integro-differential equation $$D^{2s}_xσ(x) + \int_{-∞}^{+∞}  k(x  y)σ(y)dy = g(x)$$ 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.

Xiao-Qing Jin, Vai-Kuong Sin & Jin-Yun Yuan. (1970). A Wavelet Method for the Fredholm Integro-Differential Equations with Convolution Kernel. Journal of Computational Mathematics. 17 (4). 435-440. doi:
Copy to clipboard
The citation has been copied to your clipboard