Numerical Solution of Non--Self-Adjoint Sturm--Liouville Problems and Related Systems

Abstract

This paper gives the analysis and numerics underlying a shooting method for approximating the eigenvalues of non--self-adjoint Sturm--Liouville problems. We consider even order problems with (equally divided) separated boundary conditions. The method can find the eigenvalues in a rectangle and in a left half-plane. It combines the argument principle with the compound matrix method (using the Magnus expansion). In some cases the computational cost of compound matrices can be reduced by transforming to a second order vector Sturm--Liouville problem. We study the asymptotics of the solutions of the ODE for large absolute values of the eigenvalue parameter in order to calculate the eigenvalues in a left half-plane. The method is applied to the Orr--Sommerfeld equation and other examples.