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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 \((i)\quad -y''+q(x)y=\lambda y,\) \(0\leq x\leq \pi\), \((ii)\quad y(0)=y(\pi),\) \(y'(0)=y'(\pi),\) \((iii)\quad y(0)=-y(\pi),\) \(y'(0)=-y'(\pi)\) but is not applicable for (i), (iv) or (i), (v) when \((iv)\quad y(0)=-y(\pi),\) \(y'(0)=y'(\pi),\) \((v)\quad y(0)=y(\pi),\) \(y'(0)=-y'(\pi).\) The results are confirmed by asymptotic analysis and numerical computations with \(q(x)=10\cos (2x)\) and \(q(x)=x^ 2(\pi -x).\)
Reviewer: J.B.Butler jun.

MSC:

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
34L99 Ordinary differential operators

Citations:

Zbl 0436.65063
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