Solution of a transcendental eigenvalue problem via interval analysis

A knowledge of the complex roots λ of the transcendental eigenvalue equation sinλα=±λsinα is essential in the analysis of the slow viscous fluid flow in the neighbourhood of a sharp corner which subtends an angle α ϵ (0, 2π] to the fluid. Existing methods for finding all roots λ essentially require an a priori knowledge of the solution structure; given that {λ1, λ2, …, λm} are known, glm+1 is determined via iterations, and/or a solution procedure initiated by λm.

We present herein a general interval analysis method which exhaustively finds all roots λ via only the original eigenvalue equation: no other information is required. The interval-analysis method automatically guarantees root existence and uniqueness while simultaneously providing error bounds.