A Frobenius–Euler Polynomial Matrix‐Collocation Method for Nonlinear Duffing‐Type Oscillators: Structural Conditioning, Spectral Convergence, and Piecewise Long‐Time Integration

Abstract

This paper develops a matrix‐collocation method based on the Frobenius–Euler polynomial family Hn(λ;x)${{H}_n}( {\lambda ;x} )$ for nonlinear Duffing‐type oscillators. The Appell structure of the Frobenius–Euler basis yields a parameter‐independent subdiagonal operational derivative matrix, while the parameter λ$\lambda $ controls the conditioning of the resulting linear system through the basis evaluation. The method uses Chebyshev–Gauss–Lobatto collocation nodes on the unit interval, a Newton–Raphson scheme with analytical Jacobian and homotopy continuation in the cubic coefficient, and a piecewise extension for long‐time integration over multiple oscillation periods. A structural conditioning analysis identifies a stable parameter regime λ∈[−2,−0.5]$\lambda \in [ { - 2, - 0.5} ]$ in which the condition number of the collocation system is smaller than that of the Bernoulli operational matrix scheme by a factor of 30$\hskip.001pt 30$ – 150$\hskip.001pt 150$ at truncation orders N∈[6,14]$N\in [ {6,14} ]$ , while the achievable accuracy matches that of the Bernoulli basis to floating‐point precision in the benchmark problems considered. The mechanism behind the conditioning advantage is explained through the proximity of a removable but numerically dangerous singularity in the Frobenius–Euler generating function at λ=1$\lambda = 1$ . Numerical experiments on three standard Duffing benchmark problems establish accuracy comparable to that of the improved Taylor matrix method (ITMM), the Laplace decomposition algorithm, the quasilinearized Bessel polynomial collocation method (QBPCM), and the modified variational iteration method; the piecewise scheme with K=8$K = 8$ segments of degree N=8$N = 8$ achieves final‐time errors below 5×10−8$5 \times {{10}^{ - 8}}$ over four oscillation periods of the unforced cubic Duffing oscillator.