Differential Eigenvalue Problems
This chapter utilises the finite element method in space to replace the differential eigenvalue problem by an algebraic one whose solution provides approximations to the eigenvalues and eigenfunctions. The eigenvalues may be the natural frequencies of structures, critical loads of columns, or complex frequencies of disturbance waves superimposed on an otherwise laminar fluid motion. Differential eigenvalue problems are encountered in the analysis of free vibration and stability of structures and fluid motion, among other fields in engineering sciences. Analysis of the stability of fluid motions is an important branch of fluid mechanics. If a laminar motion is subjected to small disturbances, different physical mechanisms may transfer energy from the basic laminar motion to the disturbances. And if the disturbances grow in space or time, the laminar motion can take a different form or may even lose its organized structure and become turbulent.