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A modified difference scheme for periodic and semiperiodic Sturm- Liouville problems. (English) Zbl 0834.65075
The computation of eigenvalues ${\lambda }_{1}<{\lambda }_{2}\cdots$ of the regular Sturm-Liouville problem $-{y}^{\text{'}\text{'}}+q\left(x\right)y=\lambda y$, with periodic $y\left(0\right)=y\left(\pi \right)$, ${y}^{\text{'}}\left(0\right)={y}^{\text{'}}\left(\pi \right)$ or semiperiodic $y\left(0\right)=-y\left(\pi \right)$, ${y}^{\text{'}}\left(0\right)=-{y}^{\text{'}}\left(\pi \right)$ boundary conditions using finite difference and finite element methods is considered. The reduction of the error of the eigenvalues by a simple step-dependent linear multistep method is studied. The classical finite difference method introduced by A. I. Andrew [BIT 28, No. 2, 254-269 (1988; Zbl 0646.65070)] is commented. Numerical experiments taking $q\left(x\right)={x}^{2}\left(\pi -x\right)$ in the above problem are carried out and the results are tabulated. The usefulness of the approach is shown even for low-lying eigenvalues.
Reviewer: V.Burjan (Praha)

##### MSC:
 65L15 Eigenvalue problems for ODE (numerical methods) 65L12 Finite difference methods for ODE (numerical methods) 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators