Eigenvalues of Sturm-Liouville problems with discontinuity conditions inside a finite interval.

*(English)*Zbl 1124.65066The author considers an eigenvalue problem of Sturm-Liouville type, where a discontinuity of the solution is specified at one point inside the domain of the two-point boundary value problem. For a detailed analysis, the solution is split into a left and a right part with respect to the discontinuity. Bounds related to each part are proved. A regularised sampling method has been introduced for Sturm-Liouville problems with parameter-dependent boundary conditions in the previous work by the author [Math. Comput. 74, No. 252, 1793–1801 (2005; Zbl 1080.34010)]. This method represents an improvement of techniques based on Shannon’s sampling theory, since any integration is avoided. Thus high-order approximations are achieved at low costs.

In the paper at hand, this approach is applied to the Sturm-Liouville problem with discontinuity. Used functions of a Paley-Wiener space exhibit a representation by a specific series due to the Whitaker-Shannon-Kotel’nikov theorem. A bound on the truncation error of a finite series can be given. Since the exact eigenvalues represent the squares of the zeros of some characteristic function, an approximation is obtained by using the truncated series in this function. Consequently, the author shows a bound on the approximation error of the constructed method. Finally, a brief numerical simulation of a test problem is presented, which verifies that high-order approximations are achieved.

In the paper at hand, this approach is applied to the Sturm-Liouville problem with discontinuity. Used functions of a Paley-Wiener space exhibit a representation by a specific series due to the Whitaker-Shannon-Kotel’nikov theorem. A bound on the truncation error of a finite series can be given. Since the exact eigenvalues represent the squares of the zeros of some characteristic function, an approximation is obtained by using the truncated series in this function. Consequently, the author shows a bound on the approximation error of the constructed method. Finally, a brief numerical simulation of a test problem is presented, which verifies that high-order approximations are achieved.

Reviewer: Roland Pulch (Wuppertal)

##### MSC:

65L15 | Numerical solution of eigenvalue problems involving ordinary differential equations |

65L70 | Error bounds for numerical methods for ordinary differential equations |

34B24 | Sturm-Liouville theory |

34L16 | Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators |

##### Keywords:

Sturm-Liouville problem; discontinuity condition; Shannon’s sampling theory; regularised sampling method; Whittaker-Shannon-Kotel’nikov theorem; error bound; numerical examples; two-point boundary value problem; eigenvalues
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\textit{B. Chanane}, Appl. Math. Comput. 188, No. 2, 1725--1732 (2007; Zbl 1124.65066)

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##### References:

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