For a given sensitivity parameter lambda>0, determine a function u∈ C^2([0,1]) satisfying the nonlinear boundary-value problem u''(x)=lambda sinh(lambda u(x)) for 0<x<1, with boundary conditions u(0)=0 and u(1)=1.

Troesch problem

The Troesch problem concerns a stiff nonlinear two-point boundary-value problem, commonly represented by $u''(x)=\lambda\sinh(\lambda u(x))$ with $u(0)=0$ and $u(1)=1$ . Public work has focused on numerical and asymptotic approximations rather than an exact general solution.

Known results

SI-method computations showed convergence and favorable accuracy, while leaving general existence and algorithmic convergence analysis open (2016).
Bessel-polynomial collocation methods provided error analysis and numerical agreement with established schemes (2021).
Transformation, unified-difference, quasi-quadratization, and envelope methods improved numerical treatment over tested parameter ranges (2021–2023), without resolving the problem.

August 2026 boundary-corrected approximation

Helmi Temimi and Marwan Alquran published A Boundary-Corrected Analytical Approximation for the Large-Parameter Troesch Problem. It introduces an explicit boundary correction for large-parameter boundary layers; the retrieved record gives no quantitative comparison or proof of exact resolution, so the advance remains unverified.

Current status (as of August 2026): Numerical and analytical approximations are available, including a newly published boundary-corrected approximation, but no exact general solution or proof resolving the Troesch problem has been verified.

Troesch problem

Primary source

Additional references

Progress summary

Solutions 0