Eight perspectives on the exponentially ill-conditioned equation \(\varepsilon y'' - x y' + y = 0\). (English) Zbl 1437.65075
Summary: 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).
MSC:
| 65L08 | Numerical solution of ill-posed problems involving ordinary differential equations |
| 34E20 | Singular perturbations, turning point theory, WKB methods for ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
Keywords:
ill-conditioning; boundary layer analysis; slow manifold; turning point; pseudospectra; Sturm-Liouville operatorReferences:
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