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Correction of finite difference eigenvalues of periodic Sturm-Liouville problems. (English) Zbl 0676.65089
The known error in computation of higher eigenvalues in the Sturm- Liouville problem with constant potential function q is used to improve the accuracy of approximations for non-constant q which are obtained by the centered finite difference method with uniform mesh for problems with non-separated boundary conditions. The result is known for separated boundary conditions [cf. J. W. Paine, F. R. de Hoog and R. S. Anderssen, Computing 26, 123-139 (1981; Zbl 0436.65063)]. The author proves that the method is applicable to the boundary value problems (i), (ii) and (i), (iii) where (i)-y '' +q(x)y=λy, 0xπ, (ii)y(0)=y(π), y ' (0)=y ' (π), (iii)y(0)=-y(π), y ' (0)=-y ' (π) but is not applicable for (i), (iv) or (i), (v) when (iv)y(0)=-y(π), y ' (0)=y ' (π), (v)y(0)=y(π), y ' (0)=-y ' (π)· The results are confirmed by asymptotic analysis and numerical computations with q(x)=10cos(2x) and q(x)=x 2 (π-x)·
Reviewer: J.B.Butler jun.

MSC:
65L15Eigenvalue problems for ODE (numerical methods)
34L99Ordinary differential operators