This Mathieu Functions Toolbox is used to solve Mathieu function numerically.
The Mathieu equation is a second-order homogeneous linear differential equation
and appears in several different situations in Physics: electromagnetic or
elastic
wave equations with elliptical boundary conditions as in waveguides
or resonators, the motion of particles in alternated-gradient
focussing or electromagnetic traps, the inverted pendulum,
parametric oscillators, the motion of a quantum particle in
a periodic potential, the eigenfunctions of the quantum pendulum, are just few
examples.
Their solutions, known as Mathieu functions,
were first discussed by Mathieu in 1868 in the context of the
free oscillations of an elliptic membrane.
The toolbox has 8 demos.
All demos can be acessed from "?->Scilab Demonstrations"
Features
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* mathieu_cart2ell : Converts from cartesian to elliptical coordinates.
* mathieu_ell2cart : Converts from elliptical to cartesian coordinates.
* mathieu_ellipse : 3D plot of a mode with elliptical boundary conditions.
* mathieu_elliptical : 3D plot of a mode with elliptical boundary conditions.
* mathieu_expqa : Plot nu with respect to q and a.
* mathieu_mathieu: Evaluates periodic Mathieu functions.
* mathieu_mathieuS: Calculates solutions of Mathieu Equation.
* mathieu_mathieuexp: Evaluates the characteristic exponent.
* mathieu_mathieuf: Evaluates characteristic values and expansion
coefficients.
Bibliography
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* R. Coïsson, G. Vernizzi and X.K. Yang, "Mathieu functions and numerical
solutions of the Mathieu equation", IEEE Proceedings of OSSC2009, online at http://www.fis.unipr.it/~coisson/Mathieu.pdf
* N.W. McLachlan, Theory and Application of Mathieu Functions, Oxford Univ.
Press, 1947
* J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S.
Chávez-Cerda, "Mathieu functions, a visual approach", American Journal of
Physics, 71 (233), 233-242. An introduction to applications, online at http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/modern/matheiu0.pdf
