(untitled) (Journal of Computational Mathematics, 1999)
We study the Fredholm integro-differential equation D^2s_xσ(x) + _-∞^+∞ 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 log n) operations.