Fractional Mathieu Equation

Abstract

After reviewing the concept of fractional derivative, we derive expressions for the transition curves separating regions of stability from regions of instability in the ODE: x″+(δ+εcost)x+cDαx=0 where Dαx is the order α derivative of x(t), where 0 < α < 1. We use the method of harmonic balance and obtain both a lowest order approximation as well as a higher order approximation for the n = 1 transition curves. We also obtain an expression for the n = 0 transition curves.

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

“…Fractional calculus and fractional order differential equations clearly has wide applications in areas like control theory, diffusion, heat conduction, viscoelasticity and electromagnetics. References [1,[4][5][6] cites numerous works where such applications were discussed. Many fractional differential equations have been extensively discussed and treated recently.…”

“…Many fractional differential equations have been extensively discussed and treated recently. We can find a detailed list of such equations in Rand, et al [5]. On the other hand, though much work have not been done in developing a complete theory of multi-order fractional differential equations, few literatures exist in this regard (see [7][8][9][10]).…”

“…The aim of this present work is first to, propose a more general form of the damped fractional Mathieu equation with external periodic forcing (6) and following [5] discuss the effect of the external forcing on the transition curves separating regions of stability from regions of instability in (10), secondly, to obtain an approximate analytical solution to the proposed MFDE (6) and the multi-order fractionally damped and forced Duffing-Mathieu equation using the HPM [14][15][16][17][18][19][20][21]. k , the order of the operation is a positive real number and p is an integer that satisfies…”

“…Since its introduction, extensive studies have been conducted on/with (1), these studies have revealed the importance of the equation (1) and many of its variation in nonlinear dynamics of real physical systems of engineering and other areas. Reference [1,3] presented a survey of some of the nonlinear variations of (1), some of which were listed below in (2)- (5). As was noted in Rand [4] Eq.…”

“…A detailed analysis of Mathieu's equation especially that of stability, can also be found in Rand [4]. In all these extensive studies, we note that the fractional aspect of Mathieu's equation is not left out, see [1,[4][5][6].…”