Shibata, Tetsutaro Three-term spectral asymptotics for the perturbed simple pendulum problems. (English) Zbl 1164.34450 Differ. Integral Equ. 17, No. 1-2, 215-226 (2004). Summary: This paper is concerned with the perturbed simple pendulum problem \[ -u''(t)+g(u(t))=\lambda \sin u(t),\quad u(t)>0,\quad t\in I:=(-T,T),\quad u(\pm T)=0, \] where \(\lambda >0\) is a parameter. It is known that if \(\lambda \gg 1\), then the corresponding solution develops boundary layers. We adopt a new parameter \(\epsilon \in (0,T)\) which characterizes both the boundary layers and the height of the solution, and parametrize a solution pair \((\lambda ,u)\) by \(\epsilon \), namely \((\lambda ,u)=(\lambda (\epsilon ),u_\epsilon )\), thus establishing a three-term asymptotics for \(\lambda (\epsilon )\) as \(\epsilon \to 0\). MSC: 34E05 Asymptotic expansions of solutions to ordinary differential equations 34E10 Perturbations, asymptotics of solutions to ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:perturbed simple pendulum problem; height of the solution × Cite Format Result Cite Review PDF