Abstract
When we work with differential equation modeling of a physical phenomenon, the equation usually has no closed form solution expressible by elementary functions. Engineers and scientists need to understand the behavior of the phenomenon and wish to extract as much information as possible from the equations. In this chapter, we introduce and review several approximation methods by focusing on one of the most important differential equations for which we do not have elementary solution but still we need to understand its behavior with very good approximation. We use the Mathieu equation to be used as a base to introduce some approximate methods that are useful in stability analysis of parametric differential equations. The Mathieu equation is the simplest parametric equation that directly or indirectly appears in stability analysis of dynamic systems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arscott, F. M. (1964). Periodic differential equations: An introduction to Mathieu, lame, and allied functions. Poland: Pergamon Press.
Bellman, R. E. (1970). Methods of nonlinear analysis. New York: Academic Press.
Bellman, R. E., & Kalaba, R. E. (1965). Quasilinearization and nonlinear boundary-value problems. New York: Elsevier.
Esmailzadeh, E., Mehri, B., & Jazar, R. N. (1996). Periodic solution of a second order, autonomous, nonlinear system. Journal of Nonlinear Dynamics, 10(4), 307–316.
Esmailzadeh, E., & Jazar, R. N. (1997). Periodic solution of a Mathieu-Duffing type equation. International Journal of Nonlinear Mechanics, 32(5), 905–912.
Esmailzadeh, E., & Jazar, R. N. (1998). Periodic behavior of a cantilever with end mass subjected to harmonic base excitation. International Journal of Nonlinear Mechanics, 33(4), 567–577.
Mathieu, E. (1868). Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique. Journal de Mathématiques Pures et Appliquées, 13, 137–203.
Markushevich, A. I. (1983). Recursive sequences. Moscow: Mir Publishers.
McLachlan, N. W. (1947). Theory and application of Mathieu functions. Oxford, UK: Clarendon Press.
McLachlan, N. W. (1956). Ordinary non-linear differential equations in engineering and physical sciences, 2nd edn. Oxford University Press.
Richards, J. A. (1983). Analysis of periodically time-varying systems. Heidelberg, Germany: Springer-Verlag Berlin.
Simmons, G. F. (1991). Differential equations with applications and historical notes (2nd ed.). New York: McGraw-Hill.
Starzhinskii, V. (1980). Applied Methods in the Theory of Nonlinear Oscillations. Moscow: Mir Publishers.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Science+Business Media, LLC, part of Springer Nature
About this chapter
Cite this chapter
N. Jazar, R. (2020). Mathieu Equation. In: Approximation Methods in Science and Engineering. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-0480-9_5
Download citation
DOI: https://doi.org/10.1007/978-1-0716-0480-9_5
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-0716-0478-6
Online ISBN: 978-1-0716-0480-9
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)