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

Journal article

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).

Authors: Trefethen, LN

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Review
• Volume: 62
• Issue: 2
• Pages: 439–462
• Publication date: 2020-05-07
• Acceptance date: 2019-01-04
• DOI: 10.1137/18M121232X
• EISSN: 1095-7200
• ISSN: 0036-1445
• Language: English
• Keywords: turning point; ill-conditioning; boundary layer analysis; Sturm–Liouville operator; slow manifold; pseudospectra