Geodesic
On the computing of the eigen\-values of the Orr--Sommerfeld problem
Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 1, pp. 301-305

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The paper deals with the Orr–Sommerfeld problem \begin{align*} {} \{(iR)^{-1}M^2-\alpha[q(x)M-q''(x)]\}y=-\lambda My,\\ (\pm 1)=y'(\pm1)=0, \end{align*} where $M=d^2/dx^2-\alpha^2$, $q(x)$ is the velocity profile, $R$ and $\alpha$ are Reynolds and wave numbers, respectively. We approve the Galerkin method to compute the eigenvalues of this problem provided that the basis for the method consists of the eigenfunctions of the operator $M^2$. 
@article{FPM_2002_8_1_a22,
     author = {M. I. Neiman-Zade and A. A. Shkalikov},
     title = {On the computing of the eigen\-values of the {Orr--Sommerfeld} problem},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {301--305},
     publisher = {mathdoc},
     volume = {8},
     number = {1},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2002_8_1_a22/}
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M. I. Neiman-Zade; A. A. Shkalikov. On the computing of the eigen\-values of the Orr--Sommerfeld problem. Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 1, pp. 301-305. http://geodesic.mathdoc.fr/item/FPM_2002_8_1_a22/