A Frobenius–Euler Polynomial Matrix-Collocation Method for Nonlinear Duffing-Type Oscillators: Structural Conditioning, Spectral Convergence, and Piecewise Long-Time Integration
ZAMM ZEITSCHRIFT FÜR ANGEWANDTE MATHEMATIK UND MECHANIK, cilt.12, sa.45, ss.10-40, 2026 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 12 Sayı: 45
- Basım Tarihi: 2026
- Doi Numarası: 10.1002/zamm.70550
- Dergi Adı: ZAMM ZEITSCHRIFT FÜR ANGEWANDTE MATHEMATIK UND MECHANIK
- Derginin Tarandığı İndeksler: Applied Science & Technology Source, Academic Search Ultimate (EBSCO), Engineering Source (EBSCO), Scopus, Materials Science & Engineering Collection (ProQuest), Technology Collection (ProQuest), Aerospace Database, Science Citation Index Expanded (SCI-EXPANDED), Compendex, INSPEC, MathSciNet, zbMATH
- Sayfa Sayıları: ss.10-40
- Dokuz Eylül Üniversitesi Adresli: Evet
Özet
This paper develops a
matrix-collocation method based on the Frobenius–Euler polynomial family 
for nonlinear Duffing-type oscillators. The
Appell structure of the Frobenius–Euler basis yields a parameter-independent
subdiagonal operational derivative matrix, while the parameter 
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 
in which the condition number of the
collocation system is smaller than that of the Bernoulli operational matrix
scheme by a factor of 
to 
at truncation orders 
, 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 
. Numerical experiments on three
standard Duffing benchmark problems establish accuracy comparable to that of
the improved Taylor matrix method, the Laplace decomposition algorithm, the
quasilinearized Bessel polynomial collocation method, and the modified
variational iteration method; the piecewise scheme with 
segments of degree 
achieves final-time errors below 
over four oscillation periods of the unforced
cubic Duffing oscillator.