Oscillation Theory and Numerical Solution of Fourth Order Sturm-Liouville Problems

A shooting method is developed to approximate the eigenvalues and eigenfunctions of a 4th order Sturm-Liouville problem. The main tool is a miss-distance function M(), which counts the number of eigenvalues less than. The method approximates the coeecients of the diierential equation by piecewise-constant functions, which enables an exact solution to be found on each mesh interval. In order to calculate M() for the approximate problem, certain oscillation numbers NL and NR must be computed. These consist of sums of nullities (or rank deeciencies) of 22 matrices obtained from the solutions of the approximate diierential equation. Although these solutions can be found explicitly, the calculation of NL and NR is nontrivial, and is obtained by using certain properties of M().