An Engineering Boundary Eigenvalue Problem Studied by Functional-Analytic Methods
DOI:
https://doi.org/10.24297/jam.v23i.9574Keywords:
Compact operator with simple eigenvalues, Boundary eigenvalue problem, Boundary value problem, Expansion in series of eigenvectors, Generalized Rayleigh-Quotient, Real parts of eigenvaluesAbstract
In this paper, we take up a boundary value problem (BVP) from the area of engineering that is described in a book by L. Collatz. Whereas there, the BVP is cast into a boundary eigenvalue problem (BEVP) having complex eigenvalues, here the original BVP is transformed into a BEVP that has positive simple eigenvalues and real eigenfunctions. Further, unlike there, we derive the inverse T = G of the differential operator L associated with the BEVP, show that T = G is compact in an appropriate real Hilbert space H, expand T u = Gu and u for all u ∈ H in a respective series of eigenvectors, and obtain max-, min-, min-max, and max-min-Rayleigh-quotient representation formulas of the eigenvalues. Specific examples for generalized Rayleigh quotients illustrate the theoretical findings. The style of the paper is expository in order to address a large readership.
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