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

A shooting method is developed to approximate the eigenvalues and eigenfunctions of a fourth-order Sturm-Liouville problem. The main tool is a miss-distance function M(A), which counts the number of eigenvalues less than λ. The method approximates the coefficients of the differential equation by piecewise-constant functions, which enables an exact solution to be found on each mesh interval. In order to calculate M(A) for the approximate problem, certain oscillation numbers N L and N R must be computed. These consist of sums of nullities (or rank deficiencies) of 2x2 matrices obtained from the solutions of the approximate differential equation. Although these solutions can be found explicitly, the calculation of N L and N R is non-trivial, and is obtained by using certain properties of M(A).

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