On a shooting algorithm for Sturm-Liouville eigenvalue problems with periodic and semi-periodic boundary conditions
The eigenvalues of a regular Sturm-Liouville system on \((0,a)\) with periodic and semi-periodic boundary conditions satisfy an equation (i) \(\varphi_ 1 (a, \lambda) + \varphi_ 2' (a, \lambda) = D (\lambda) = \pm 2\), where \(\varphi_ 1\), \(\varphi_ 2\) are basic solutions of the differential equation satisfying initial conditions \(\varphi_ 1 (0, \lambda) = 1\), \(\varphi_ 1' (0, \lambda) = 0\), \(\varphi_ 2 (0, \lambda) = 0\), \(\varphi_ 2' (0, \lambda) = 1\). The author applies the shooting method to generate a sequence \(\lambda^{(k)}\), \(\mu^{(k)}\) of numbers and functions \(\varphi_ i (x, \lambda^{(k)})\), \(\varphi_ i (x, \mu^{(k)})\), \(i=1,2\), such that \(\lambda^{(k)}\), \(\mu^{(k)}\) converge to eigenvalues \(\lambda, \mu\) of the periodic and semi-periodic boundary value problems respectively. Starting values for \(\lambda^{(k)}\), \(\mu^{(k)}\) are based on the known behavior of \(D(\lambda)\). The new algorithm is applied to three test problems, one of which is the Mathieu equation. Calculations made on a CYBER 860 computer for the first ten pairs of eigenvalues are accurate to \(10^{-4}\).
scientific article; zbMATH DE number 569522