Physical conditions in the analytical solution on the Mathieu equation
Condiciones físicas en la solución analítica de la ecuación de Mathieu
DOI:
https://doi.org/10.56712/latam.v5i6.3112Palabras clave:
boundary problems, mathieu equation, hill equation, perturbation theoryResumen
The study aims to analyze the Mathieu equation associated with linear systems under properties, use the one-dimensional Floquet theorem and explain the behavior of the solutions of the Mathieu equation and function relating to the Hill equation together with the theory of disturbances. The most common mathematical problem in physics is finding solutions to certain second-order differential equations subject to boundary conditions. A hypothetical deductive approach method, handling qualitative equations of mathematical physics, was adopted in review. With Mathieu's equation, the existence of the physical conditions that influence the value of the parameters that define the equation were answered. It is important due to its usefulness in the field of analytical and mechanical dynamics, physical systems subjected to parametric excitation; In addition, it leaves other variations open for new applications.
Descargas
Citas
Arnold, V. I. (1973). Ordinary Differential Equations. MIT Press.
Barrow, G. J. (1996). Poincaré and the Three Body Problem. American Mathematical Society. https://oro.open.ac.uk/57403/1/335423.pdf
Bickley, W. G. (1940). A class of hyperbolic Mathieu functions. Philosophical Magazine, 30, 312. https://doi.org/10.1080/14786444008520720
Cesari, L. (1963). Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. Academy Press.
Edwards, H. and Penney, D. (2000). Elementary Differential Equations with Boundary Value Problems. Pretince Hall.
Feynman, R. P. and Hibbs, A. R. (1965). Quantum Mechanics and Path Integrals. McGraw-Hill. https://books.google.com.pe/books?id=JkMuDAAAQBAJ&printsec=frontcover&hl=es#v=onepage&q&f=false
Hartman, P. (1964). Ordinary Differential Equations. Jhon Wiley.
Hille, E. (1923). Zeros of the Mathieu function. Proceedings London Mathematical Society. https://doi.org/10.1112/plms/s2-23.1.185
Hirsch, M. W. and Smale, S. (1974). Differential Equations, Dynamical System and Linear Algebra. Academy Press. file:///C:/Users/Borda/Downloads/TheworkofStephenSmaleindifferentialtopology.pdf
Jeffreys, H. (1924). Certain Solutions of Mathieu´s equations. Proceedings London Mathematical Society, 23, 449.
Jordan, D. W. and Smith, P. (1999). Nonlinear Ordinary Differential Equations. Clarendon Press Oxford Applied and Engineering Mathematics.
Kreyszig, E. and Normington, E. J. (1994). Maple Computer Manual for Advanced Engineering Mathematics. Wiley, New-York.
Marshall, W. (1909). Asymptotic representation of elliptic cylinder functions. Inaugural Dissertation, Zurich.
Nayfeh, A. H. and Mook, D. T. (1995). Nonlinear oscillations. New York, NY: Wiley Interscience.
Naytch, A. (1973). Perturbation Methods. Wiley y Sons Inc., Ney York.
Richards, D. (2002). Advanced Mathematical Methods With Maple. Cambridge University Press. https://catdir.loc.gov/catdir/samples/cam031/2001025976.pdf
Ross, C. (1995). Differential Equations: An Introduction with Mathematical. Springer-Verlag, New York.
Shingareva, I. (1995). Investigation of Standing Surface Waves in a Fluid of Finite Depth by Computer Algebra Methods. PhD thesis, Institute for Problems in Mechanics, RAS, Moscow.
Vvedensky, D. (1993). (1993). Partial Differential Equations with Mathematica. Adison-Wesley.
Whittaker, E. T. (1914). General solution of Mathieu´s equation. Proceedings Edinburgh Mathematica Society, 32, 75.
Young, A. W. (1914). Quasi-periodic solutions of Mathieu´s equation. Proceedings Edinburgh Mathematical Society, 32, 81.
Zimmerman, Robert L. and Olness, Frederick. (1995). Mathematica for Physicists. Addison-Wesley. https://www.abebooks.com/9780201537963/Mathematicar-Physics-Robert-Zimmerman-Fredrick-0201537966/plp
