Analytical Approximant to a Quadratically Damped Forced Cubic-Quintic Duffing Oscillator.

The cubic-quintic Duffing oscillator of a system with strong quadratic damping and forcing is considered. We give elementary approximate analytical solution to this oscillator in terms of exponential and trigonometric functions. We compare the analytical approximant with the Runge-Kutta numerical solution. The approximant allows us to estimate the points at which the solution crosses the horizontal axis.

The Galerkin Method for Solving Strongly Nonlinear Oscillators.

In this paper, we make use of the Galerkin method for solving nonlinear second-order ODEs that are related to some strongly nonlinear oscillators arising in physics and engineering. We derive the iterative schemes for finding the coefficients that appear in the linear Galerkin hat combination in the ansatz form solution. These coefficients may be found iteratively by solving either a quadratic or a higher degree algebraic equation. Examples are presented to illustrate the obtained results. Some exact solutions are given, and they are compared with both the Runge-Kutta numerical solution and the solution obtained using the Galerkin finite element method.

Solution to a Damped Duffing Equation Using He's Frequency Approach.

In this paper, we generalize He's frequency approach for solving the damped Duffing equation by introducing a time varying amplitude. We also solve this equation by means of the homotopy method and the Lindstedt-Poincaré method. High accurate formulas for approximating the Jacobi elliptic function cn are formally derived using Chebyshev and Pade approximation techniques.