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**Numerical solution of non–self-adjoint Sturm–Liouville problems and related systems.**
*(English)*
Zbl 1001.34020

For selfadjoint regular and singular Sturm-Liouville problems, there are various established methods to calculate the (real) spectrum numerically. For nonselfadjoint problems, other methods have to be used, and the most powerful ones are those taking advantage of the rich structure of analytic functions. Here, such an approach is thorouhgly investigated, starting with a theoretic result on higher-order Sturm-Liouville problems with separated boundary conditions and leading to numerical results for a number of examples. Using the shooting method, the miss-distance function \(f(\lambda)\) is introduced, and \(\lambda \) is an eigenvalue if and only if \(f(\lambda)=0\). The argument principle
\[
N(f,\Gamma)=\frac 1{2\pi i}\int_\Gamma f'(\lambda)f(\lambda)^{-1} d\lambda \tag{1}
\]
is used to calculate the number \(N(f,\Gamma)\) of zeros of \(f\) inside the curve \(\Gamma \). Particular attention is given to the left half-plane since the knowledge of the location of eigenvalues therein is important for stability. For an effective implementation of the numerical code, a thorough investigation of the \(n\)th-order differential equation with \(n\) even has to be done. In particular, it is shown that \(f(\lambda)\) tends to a nonzero number as \(|\lambda |\to\infty \), Re \(\lambda \leq 0\), which implies that (1) holds with \(\Gamma \) being the imaginary axis and \(N(f,\Gamma)\) the number of eigenvalues in the left half-plane.

The Orr-Sommerfeld equation with constant and \(\lambda \)-dependent boundary conditions is investigated explicitly. For numerical implementations, the compound matrix method, the Magnus expansion, and the vector Sturm-Liouville method are used.

Six examples with tables of numerical results complete the paper, where the Magnus and vector SL method are compared. One of the examples numerically confirms the stability/instability boundary predicted by C. I. Gheorghiu and I. S. Pop [Stancu, Dimitrie D. (ed.) et al., Approximation and optimization, Proceedings of ICAOR: International conference, Cluj-Napoca, Romania, July 29–August 1, 1996. Volume II. Cluj-Napoca: Transilvania Press, 119-126 (1997; Zbl 0888.76065)] for the Orr-Sommerfeld equation for a liquid film flowing over an inclined plane.

The Orr-Sommerfeld equation with constant and \(\lambda \)-dependent boundary conditions is investigated explicitly. For numerical implementations, the compound matrix method, the Magnus expansion, and the vector Sturm-Liouville method are used.

Six examples with tables of numerical results complete the paper, where the Magnus and vector SL method are compared. One of the examples numerically confirms the stability/instability boundary predicted by C. I. Gheorghiu and I. S. Pop [Stancu, Dimitrie D. (ed.) et al., Approximation and optimization, Proceedings of ICAOR: International conference, Cluj-Napoca, Romania, July 29–August 1, 1996. Volume II. Cluj-Napoca: Transilvania Press, 119-126 (1997; Zbl 0888.76065)] for the Orr-Sommerfeld equation for a liquid film flowing over an inclined plane.

Reviewer: Manfred MĂ¶ller (Johannesburg)

### MSC:

34B24 | Sturm-Liouville theory |

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

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

65L20 | Stability and convergence of numerical methods for ordinary differential equations |