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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 $\left(i\right)\phantom{\rule{1.em}{0ex}}-{y}^{\text{'}\text{'}}+q\left(x\right)y=\lambda y,$ $0\le x\le \pi$, $\left(ii\right)\phantom{\rule{1.em}{0ex}}y\left(0\right)=y\left(\pi \right),$ ${y}^{\text{'}}\left(0\right)={y}^{\text{'}}\left(\pi \right),$ $\left(iii\right)\phantom{\rule{1.em}{0ex}}y\left(0\right)=-y\left(\pi \right),$ ${y}^{\text{'}}\left(0\right)=-{y}^{\text{'}}\left(\pi \right)$ but is not applicable for (i), (iv) or (i), (v) when $\left(iv\right)\phantom{\rule{1.em}{0ex}}y\left(0\right)=-y\left(\pi \right),$ ${y}^{\text{'}}\left(0\right)={y}^{\text{'}}\left(\pi \right),$ $\left(v\right)\phantom{\rule{1.em}{0ex}}y\left(0\right)=y\left(\pi \right),$ ${y}^{\text{'}}\left(0\right)=-{y}^{\text{'}}\left(\pi \right)·$ The results are confirmed by asymptotic analysis and numerical computations with $q\left(x\right)=10cos\left(2x\right)$ and $q\left(x\right)={x}^{2}\left(\pi -x\right)·$
Reviewer: J.B.Butler jun.

##### MSC:
 65L15 Eigenvalue problems for ODE (numerical methods) 34L99 Ordinary differential operators