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A modified difference scheme for periodic and semiperiodic Sturm- Liouville problems. (English) Zbl 0834.65075
The computation of eigenvalues λ 1 <λ 2 of the regular Sturm-Liouville problem -y '' +q(x)y=λy, with periodic y(0)=y(π), y ' (0)=y ' (π) or semiperiodic y(0)=-y(π), y ' (0)=-y ' (π) 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(x)=x 2 (π-x) 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:
65L15Eigenvalue problems for ODE (numerical methods)
65L12Finite difference methods for ODE (numerical methods)
65L06Multistep, Runge-Kutta, and extrapolation methods
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators