A WAVELET METHOD FOR THE FIRST KIND INTEGRAL EQUATIONS WITH KERNEL $k(x-y)$

Abstract

We study the first kind integral equation $$ \displaystyle\int^{+\infty}_0 k(x-y)\sigma (y)dy = g(x) $$ by the wavelet method. The integral equation is discretized with respect to two different wavelet bases. We then have two different linear systems. One of them is a Toeplitz system and the other one is a system with condition number $\kappa =O(1)$ after a diagonal scaling. By using the preconditioned conjugate gradient (PCG) method with the fast wavelet transform (FWT) and the fast Fourier transform (FFT), we can solve the systems in $O(n \log n)$ operations where $n$ is the size of the systems.