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Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics

Brown, B. Malcolm and Langer, M. and Marletta, M. and Tretter, C. and Wagenhofer, M. (2010) Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics. LMS Journal of Computation and Mathematics, 13. pp. 65-81. ISSN 1461-1570

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In this paper we present computer-assisted proofs of a number of results in theoretical fluid dynamics and in quantum mechanics. An algorithm based on interval arithmetic yields provably correct eigenvalue enclosures and exclosures for non-self-adjoint boundary eigenvalue problems, the eigenvalues of which are highly sensitive to perturbations. We apply the algorithm to: the Orr-Sommerfeld equation with Poiseuille profile to prove the existence of an eigenvalue in the classically unstable region for Reynolds number R=5772.221818; the Orr-Sommerfeld equation with Couette profile to prove upper bounds for the imaginary parts of all eigenvalues for fixed R and wave number α; the problem of natural oscillations of an incompressible inviscid fluid in the neighbourhood of an elliptical flow to obtain information about the unstable part of the spectrum off the imaginary axis; Squire's problem from hydrodynamics; and resonances of one-dimensional Schrödinger operators.