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This study introduces an iterative Galerkin method to solve nonlinear second-order ordinary differential equations (ODEs) common in physics and engineering. The approach provides accurate solutions for strongly nonlinear oscillators, validated against numerical and finite element methods.

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Area of Science:

  • Applied Mathematics
  • Computational Physics
  • Engineering Mechanics

Background:

  • Nonlinear second-order ordinary differential equations (ODEs) model complex phenomena in physics and engineering.
  • Strongly nonlinear oscillators present significant analytical and numerical challenges.

Purpose of the Study:

  • To develop and present an iterative Galerkin method for solving nonlinear second-order ODEs.
  • To derive iterative schemes for determining coefficients in an ansatz-based solution.
  • To validate the method's accuracy against established numerical techniques.

Main Methods:

  • Application of the Galerkin method.
  • Derivation of iterative schemes for coefficient calculation.
  • Solving quadratic and higher-degree algebraic equations.
  • Comparison with Runge-Kutta and Galerkin finite element methods.

Main Results:

  • The proposed iterative Galerkin method yields accurate solutions for nonlinear oscillators.
  • Iterative schemes were successfully derived for coefficient determination.
  • Obtained solutions show good agreement with Runge-Kutta and Galerkin finite element results.

Conclusions:

  • The iterative Galerkin method is an effective tool for analyzing strongly nonlinear oscillators.
  • The method offers a reliable alternative for solving challenging ODEs in science and engineering.