The reducing rank method to solve third-order Duffing equation with the homotopy perturbation

In the current work, we apply a nonstandard scheme to solve the third-order Duffing equation. This equation is produced from the strong damped Klein–Gordon equation under the traveling wave transformation. The solution and the stability conditions for the third-order Duffing equation have been discussed for the first time. The present analysis is new and used the reducing rank method with the homotopy perturbation method. A nonoscillator solution with the oscillating solutions is derived individually and frequency formula is obtained. Besides, the stability analysis is discussed.