Global Error Minimization method for solving strongly nonlinear oscillator differential equations

A modified variational approach called Global Error Minimization (GEM) method is developed for obtaining an approximate closed-form analytical solution for nonlinear oscillator differential equations. The proposed method converts the nonlinear differential equation to an equivalent minimization problem. A trial solution is selected with unknown parameters. Next, the GEM method is used to solve the minimization problem and to obtain the unknown parameters. This will yield the approximate analytical solution of the nonlinear ordinary differential equations (ODEs). This approach is simple, accurate and straightforward to use in identifying the solution. To illustrate the effectiveness and convenience of the suggested procedure, a cubic Duffing equation with strong nonlinearity is considered. Comparisons are made between results obtained by the proposed GEM method, the exact solution and results from five recently published methods for addressing Duffing oscillators. The maximal relative error for the frequency obtained by the GEM method compared with the exact solution is 0.0004%, which indicates the remarkable precision of the GEM method.