stability analysis of a damped nonlinear wave equation

The current manuscript is concerned with extracting an analytical approximate periodic solution of a damped cubic nonlinear klein-gordon equation. the riemann-liouville fractional calculus is utilized to obtain an analytic approximate solution. the homotopy technique is absorbed in the multiple timespatial scales. the approved scheme yields a generalization of the homotopy equation; whereas, two different small parameters are adapted. the first parameter concerns with the temporal perturbation, simultaneously, the second one is accompanied by the spatial one. therefore, the analytic approximate solution needs the two perturbation expansions. this approach conducts more advantages in handling the classical multiple scales method. furthermore, the initial conditions are included throughout the multiple scale method to achieve a special solution of the governing equation of motion. the analysis ends up deriving two first-order equations within the extended variables and their actual solution is achieved. the procedure adopted here is very promising and powerful in managing similar numerous nonlinear problems arising in physics and engineering. furthermore, the linearized stability of the corresponding ordinary duffing differential equation is analyzed. additionally, some phase portraits are shown.