On the Numerical Solution of Fourth-Order Linear Two-Point Boundary Value Problems

This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. However, current methods of solution involve discretizing the differential equation directly by finite elements or finite differences, and consequently suffer from the poor conditioning introduced by such schemes. Our new method instead reformulates the equation as a collection of second-kind integral equations defined on local subdomains. Each such equation can be stably discretized. The boundary values of these local solutions are matched by solving a banded linear system. The method of iterative refinement is then used to increase the accuracy of the scheme. Iterative refinement requires applying the differential operator to a function on the entire domain, for which we provide an algorithm with linear cost. We illustrate the performance of our method on several numerical examples.