Numerical Solution of One-Dimensional Integral and Differential Equations.

摘要

Many problems in mathematical physics can be formulated as one-dimensional integral equations. Examples include problems in electrostatics, crack problems in elastic bodies, and two-point boundary value problems for ordinary differential equations. Since most integral equations arising in applications do not have analytic solutions, there is considerable interest in the numerical solution of these problems. Unfortunately, discretization of integral equations leads to dense systems of linear algebraic equations, and the direct solution of a dense linear system of dimension N requires order O(N cubed) arithmetic operations. Alternatively, the solution to the linear system can be obtained using an iterative method such as the conjugate gradient algorithm or the conjugate residual algorithm. This thesis is based on the observation that one-dimensional integral operators can be recursively decomposed into sums of products of operators of low numerical rank. A complicated analytical apparatus is then constructed which allows for the direct solution of an integral equation in order O(N) operations. The algorithms of this thesis permit the use of schemes with extremely high orders of convergence, and are quite insensitive to end-point singularities. The performance of the methods is illustrated with numerical examples.