A Wavelet Method for the Fredholm Integro-Differential Equations with Convolution Kernel (Journal of Computational Mathematics, 1999)
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 funtion to be found, $k(y)$ isa convolution kernel and $g(x)$ is a given function. Following the idea in [7], theequation is discretized with respect to two different wavelet bases. We then havetwo 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. Theother one is a system $(B_n + C_n)y = d$ with condition number $k = O(1)$ after adiagonal scaling. By using the preconditioned conjugate gradient (PCG) methodwith the fast wavelet transform (FWT) and the fast iterative Toeplitz solver, we can solve the systems in $O(n$ ${\rm log}$ $n)$ operations.
Keywords: Fredholm integro-differential equation; Kernel; Wavelet transform; Toeplitz matrix; Hankel matrix; Sobolev space; PCG method.