Numerical solution of integral equations with nonsmooth kernel and applications.

摘要

The Fredholm integral equation of the second kind and the Wiener-Hopf integral equation have been important tools in mathematical, physical and engineering applications ([1], [2], [3], [4], [5], and [6]). In this thesis we propose a new highly accurate numerical approximation scheme based on a Gauss type Clenshaw-Curtis quadrature for the Fredholm integral equation of the second kind whose kernel is either discontinuous or not smooth along the main diagonal. This scheme is of spectral accuracy when the kernel k(t,s) is infinitely differentiable away from the diagonal t = s, and is also applicable when k(t, s) is singular along the boundary, and at isolated points on the main diagonal. This numerical scheme is also applicable to the Wiener-Hopf integral equation whose kernel is of type k(| t − s|). The adaptive quadrature rule developed in this thesis is an efficient tool when the kernels decay not only exponentially but also quadratically. Applications to radial and integro-differential Schrödinger equations are also described.