Abstract
We study a boundary value problem for a fractional differential equation modeling the damped vibrations of thin film MEMS with variable potential. The principal differential part of the equation under consideration is the composition of left- and right-sided Caputo derivatives. We find sufficient conditions for the potential which guarantee the uniqueness and solvability of the problem under study. The condition we give has an integral form and is an analog of the Lyapunov inequality.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Rekhviashvili, S., Pskhu, A., Potapov, A., Feng, T., Eneeva, L.: Modeling Damped vibrations of thin film MEMS: fractional approach. In this Book (2019
Rekhviashvili, S.S.: The Lagrange formalism with fractional derivatives in problems of mechanics. Tech. Phys. Lett. 30(1), 55–57 (2004)
Stankovic, B.: An equation with left and right fractional derivatives, vol. 80, issue no. 94. Publications de l’institut mathematique. Nouvelle serie, pp. 259–272 (2006)
Atanackovic, T.M., Stankovic, B.: On a differential equation with left and right fractional derivatives. Fract. Calc. Appl. Anal. 10(2), 139–150 (2007)
Torres, C.: Existence of a solution for the fractional forced pendulum. J. Appl. Math. Comput. Mech. 13(1), 125–142 (2014)
Eneeva, L.M.: Boundary value problem for differential equation with fractional order derivatives with different origins. Vestnik KRAUNC. Fiz. Mat. Nauki. 11(2), 36–40 (2015)
Tokmagambetov, N., Torebek, B.T.: Fractional analogue of Sturm-Liouville operator. Doc. Math. 21, 1503–1514 (2016)
Eneeva, L.M.: An estimate for the first eigenvalue of the Dirichlet problem for an ordinary differential equation with fractional derivatives with different origins. News Kabardin-Balkar Sci. Center RAS 1(75), 34–40 (2017)
Eneeva, L.M.: On Neumann problem for equation with fractional derivatives with different starting points. Vestnik KRAUNC. Fiz. Mat. Nauki. 24(4), 61–65 (2018)
Acknowledgements
The authors prepared this article under the support of “Leading Talents of Guangdong Province”, № 00201502 (2016–2020) in the Jinan University (Guangzhou, China) and RFBR (project 18-51-45005).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Eneeva, L., Pskhu, A., Potapov, A., Feng, T., Rekhviashvili, S. (2021). Lyapunov Inequality for a Fractional Differential Equation Modeling Damped Vibrations of Thin Film MEMS. In: WU, C.H., PATNAIK, S., POPENTIU VLÃDICESCU, F., NAKAMATSU, K. (eds) Recent Developments in Intelligent Computing, Communication and Devices. ICCD 2019. Advances in Intelligent Systems and Computing, vol 1185. Springer, Singapore. https://doi.org/10.1007/978-981-15-5887-0_65
Download citation
DOI: https://doi.org/10.1007/978-981-15-5887-0_65
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-5886-3
Online ISBN: 978-981-15-5887-0
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)