# Eight perspectives on the exponentially ill-conditioned equation εy'' − xy' + y = 0

Trefethen, L

7 May 2020

SIAM Review

## Last Updated:

2020-08-10T13:04:35.653+01:00

2

62

## DOI:

10.1137/18M121232X

439–462-

## abstract:

Boundary-value problems involving the linear differential equation $\varepsilon y'' - x y' + y = 0$ have surprising properties as $\varepsilon\to 0$. We examine this equation from eight points of view, showing how it sheds light on aspects of numerical analysis (backward error analysis and ill-conditioning), asymptotics (boundary layer analysis), dynamical systems (slow manifolds), ODE theory (Sturm--Liouville operators), spectral theory (eigenvalues and pseudospectra), sensitivity analysis (adjoints and SVD), physics (ghost solutions), and PDE theory (Lewy nonexistence).

957098

Submitted

Journal Article