Oscillation theory and numerical solution of fourth-order Sturm--Liouville problems

A shooting method is developed to approximate the eigenvaluesand eigenfunc-tions of a fourth-order Sturm-Liouville problem.The main tool is a miss-distance function M(), which countsthe number of eigenvalues less than A. The method approximatesthe coefficients of the differential equation by piecewise-constantfunctions, which enables an exact solution to be found on eachmesh interval. In order to calculate N() for the approximateproblem, certain oscillation numbers N_L and N_R must be computed.These consist of sums of nullities (or rank deficiencies) of2 x 2 matrices obtained from the solutions of the approximatedifferential equation. Although these solutions can be foundexplicitly, the calculation of N_L and N_R is non-trivial, andis obtained by using certain properties of M().