Abstract The paper deals with a combined technique and variational theory for solving highly nonlinear fractional-order oscillator problems, which are frequently encountered in practical applications. These methods are applicable to both weakly and strongly nonlinear equations. Since most such systems do not have analytical solutions, various numerical solutions are especially needed in applied sciences. The fractional equation is transformed into an ordinary differential equation by utilizing a fractional complex transformation, and an effective method based on a modification of He’s frequency formulation is proposed. In this method, for two conditions in which the amplitude is either extremely small or remarkably large, the oscillators are divided into two extreme cases, and by matching these extreme conditions, a new frequency formulation is obtained. Furthermore, the nonlinear oscillators are solved using a variational approach for balance. Examples of higher-order and unconventional fractional Duffing equations are given for comparison. The dynamic behavior of these equations is extremely rich. Numerical calculations are performed for various amplitudes, and the frequency-amplitude relationship and relative errors are presented in tables and graphs. The solution procedure is simple and does not require linearization, and the results obtained are valid over the whole solution domain. Finally, the effectiveness of the modification and iteration approaches is confirmed by showing that the approximate and exact frequency results are in good agreement. In addition, the performance of the proposed models is compared with results available in the literature, demonstrating that they are effective methods in terms of computational efficiency and accuracy.
Şerife Faydaoğlu (Wed,) studied this question.
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