Combined Adaptive Solver for the Damped Amsler Equation

I present a fast, accurate numerical solver for the damped Amsler equation \(t u''+(1+α)u'=sin u\) with \(α∈(-12,0]\). The method combines a convergent Taylor series around the regular singular point \(t=0\) and an adaptive Taylor integrator in the logarithmic variable \(z=ln t\) for \(t≥1\). The adaptive step size is controlled by a robust radius estimator and a strict half‑sum convergence test. Symmetries extend the solution to the negative half‑axis. Numerical experiments show excellent agreement with NDSolve, and a finite‑difference residual of \(10⁻⁶\)–\(10⁻⁵\) confirms the accuracy of the combined approach. The complete Wolfram Language implementation is provided.