A one-step algorithm for strongly non-linear full fractional duffing equations

In the current study, a one-step numerical algorithm is presented to solve strongly non-linear full fractional duffing equations. A new fractional-order operational matrix of integration via  quasi-hat functions (QHFs) is introduced. Utilizing the operational matrices of QHFs, the main problem will be transformed into  a number of univariate polynomial equations. Absolute errors of the results in approximations and convergence analysis are addressed. Ultimately, five examples are provided to illustrate the capabilities of this algorithm. The numerical results are illustrated in some Tables and Figures, for different values of the parameters $\alpha~ and~ \beta$.