Transition curves approximation for Mathieu differential equation with two fractional derivatives

Subject Areas : Numeric Analyze

Hojjat Ghorbani ¹ , Yaghoub Mahmoudi ² , Farhad Dastmalchi Saei ³ , Mohammad Jahangiri Rad ⁴

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Received: 2021-09-23 Accepted : 2022-12-05 Published : 2024-09-25

Keywords: منحنی انتقال, مشتق کسری کاپوتو, معادله دیفرانسیل متیو, روش موازنه هارمونیک,

Abstract :

Mathieu differential equation in a linear second kind differential equation, which appears in different areas in applied mathematics and engineering. Most of the flat oscillators are modeled as Mathieu differential equation. One of the most important studies in this area is the study of transition curves and stability behavior of the solution. In this paper, the Mathieu differential equation with two fractional derivatives is studied. The transition curves, which separate the stability and the instability region in the parameters plane of the problem, are approximated using Harmonic Balance method. The Graph of parameter changes is plotted to achieve instability. Finding the optimal order of fractional derivatives to reach the maximum amplitude to initiate instability is discussed. The results show that if we convert the derivatives of the Mathieu differential equation to the integer order derivative, the results obtained from this method are consistent with the results obtained in other texts.

References:


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