Fast solution methods for fredholm integral equations of the second kind

Summary

The main purpose of this paper is to describe a fast solution method for one-dimensional Fredholm integral equations of the second kind with a smooth kernel and a non-smooth right-hand side function. Let the integral equation be defined on the interval [−1, 1]. We discretize by a Nyström method with nodes {cos(πj/N)} ^=0/N _j . This yields a linear system of algebraic equations with an (N+1)×(N+1) matrixA. GenerallyN has to be chosen fairly large in order to obtain an accurate approximate solution of the integral equation. We show by Fourier analysis thatA can be approximated well by$\mathop A\limits^ \sim$, a low-rank modification of the identity matrix. ReplacingA by$\mathop A\limits^ \sim$ in the linear system of algebraic equations yields a new linear system of equations, whose elements, and whose solution$\tilde x$, can be computed inO (N logN) arithmetic operations. If the kernel has two more derivatives than the right-hand side function, then$\tilde x$ is shown to converge optimally to the solution of the integral equation asN increases.

We also consider iterative solution of the linear system of algebraic equations. The iterative schemes use bothA andÃ. They yield the solution inO (N ²) arithmetic operations under mild restrictions on the kernel and the right-hand side function.

Finally, we discuss discretization by the Chebyshev-Galerkin method. The techniques developed for the Nyström method carry over to this discretization method, and we develop solution schemes that are faster than those previously presented in the literature. The schemes presented carry over in a straightforward manner to Fredholm integral equations of the second kind defined on a hypercube.