Multi-Order Fractional Mathieu Equation with External Multi-Periodic Excitation

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Abstract:

This paper presents an investigation of the behavior of the multi-order fractional differential equation (MFDE) represented in equation (6). We derive expressions for the transition curves separating regions of stability from instability for the MFDE generally and the particular case  . Employing the harmonic balance technique, we obtained approximate expressions for the   and   transition curves of  the MFDE above and particularly for the case  . We also obtained an approximate analytical solution to the multi-order fractionally damped and forced Duffing-Mathieu equation as well as some special cases computationally using the Homotopy Perturbation Method (HPM).

Keywords:

Homotopy perturbation method, Parametric excitation, Fractional calculus, Harmonic balancing method, Damping.