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

Abstract

This paper presents an investigation of the behavior of the multi-order fractional differential equation (MFDE). We derive expressions for the transition curves separating regions of stability from instability for the MFDE generally and the particular case K=2. Employing the harmonic balance technique, we obtained approximate expressions for the n=1 and n=2 transition curves of the MFDE and particularly for the case k=2. 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).