Finite-Difference Techniques for a Boundary Problem with an Eigenvalue in a Boundary Condition

A mixed boundary-value problem of elliptic type is considered in which the boundary conditions contain an eigenvalue. This problem is motivated by the investigation of small fluid oscillations in an axially symmetric tank. Several finite-difference procedures for determining the lowest eigenvalues are described. Two of the methods are found to be quite practical. The theoretical results include a discussion of the convergence of the eigenvalues of the difference problem to those of the differential problem, a proof that a certain iterative procedure provides a sequence converging to the fundamental eigenvalue and certain results concerning the nature of the eigenvalues.