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Studying highly nonlinear oscillators using the non-perturbative methodology.

Galal M Moatimid1, T S Amer2, A A Galal3

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A new non-perturbative methodology (NPM) simplifies analyzing strong nonlinear oscillators (NOSs) by transforming nonlinear equations into linear ones. This approach offers accurate solutions and stability analysis, outperforming traditional perturbation techniques.

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Area of Science:

  • Applied Mathematics
  • Nonlinear Dynamics
  • Engineering Mathematics

Background:

  • Nonlinear oscillators (NOSs) are prevalent in various scientific and engineering fields.
  • Traditional methods for analyzing strong NOSs often rely on perturbation techniques, which have limitations.

Purpose of the Study:

  • To introduce and examine a novel non-perturbative methodology (NPM) for analyzing strong nonlinear ordinary differential equations (ODEs).
  • To demonstrate the simplicity, efficiency, and accuracy of NPM compared to existing perturbation methods.

Main Methods:

  • The study employs the general He's frequency formula (HFF) within the NPM framework.
  • The NPM transforms nonlinear ODEs into equivalent linear ODEs, yielding new frequency and damping terms.
  • Theoretical results are verified using numerical comparisons with Mathematical Software (MS).

Main Results:

  • The NPM provides analytical representations for strong NOSs with reduced computational effort.
  • Numerical comparisons show excellent consistency between theoretical and precise numerical solutions.
  • The NPM overcomes limitations of Taylor expansion used in traditional perturbation methods.

Conclusions:

  • The non-perturbative solution (NPS) derived from NPM is a more reliable tool for analyzing strong NOSs.
  • NPM enables stability analysis, a capability lacking in older conventional approaches.
  • The NPS is versatile and applicable to a wide range of nonlinear problems in applied science and engineering, particularly dynamical systems.