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Mathieu functions >> Mathieu functions > Overview

Overview

An overview of the Mathieu Functions toolbox.

Purpose

This Mathieu Functions Toolbox is used to solve Mathieu function numerically [1].

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 canonical form of Mathieu equation is [2,3]:  (1),  

where the constants a, q are often referred as characteristic number (or eigenvalue) and parameter [4], respectively.

The modified Mathieu equation is [2,3]:   (2).

We present both the Floquet solution [1] and solution in angular and radial (modified) functions.

Coordinate transformation

Almost all tasks, involving Mathieu functions have elliptical geometry. For coordinate transformation we have functions:
Function Purpose
mathieu_cart2ell convert coordinates from Cartesian to elliptical
mathieu_ell2cart convert coordinates from elliptical to Cartesian
mathieu_cart2pol convert coordinates from Cartesian to polar
mathieu_pol2cart convert coordinates from polar to Cartesian
mathieu_ell_in_pol calculate polar coordinates of a point at known angle on ellipse

Floquet solution

For Floquet solution we have the following functions:
Function Purpose
mathieu_mathieuf evaluate characteristic values and expansion coefficients
mathieu_mathieu evaluate periodic Mathieu functions by calling mathieu_mathieuf
mathieu_mathieuexp calculate the characteristic exponent of non-periodic solutions and the coefficients of the expansion of the periodic factor
mathieu_mathieuS calculate solutions of ME with arbitrary a and q parameters (uses mathieu_mathieuexp)

For Floquet solutions we have demos:

Solution in angular and radial (modified) functions

We present solutions of Mathieu equation (1) as angular Mathieu functions [2,3]:   and   [2]

and solutions of modified Mathieu equation (2) as radial Mathieu functions [2, 3]: ,   ,   ,     and   ,   ,   ,   .

Before calculating any Mathieu function we calculate expansion coefficients and eigenvalues for given order m and parameter q using tri-diagonal matrixes [5-8].

We have the following functions for computation solutions of Mathieu equations:
Function Purpose Notation
mathieu_Arm compute expansion coefficients 'Arm' and eigenvalue 'am' for even angular and radial Mathieu functions   and  
mathieu_Brm compute expansion coefficients 'Brm' and eigenvalue 'bm' for odd angular and radial Mathieu functions   and  
mathieu_ang_ce compute even angular Mathieu function 'ce' or its first derivative   or  
mathieu_ang_se compute odd angular Mathieu function 'se' or its first derivative   or  
mathieu_rad_mc compute even radial (modified) Mathieu function 'Mc' or its first derivative (kinds 1 and 2)   or  
mathieu_rad_ms compute odd radial (modified) Mathieu function 'Ms' or its first derivative (kinds 1 and 2)   or  
mathieu_rad_ce compute even radial (modified) Mathieu function of the first kind 'Ce' or its first derivative   or  
mathieu_rad_se compute odd radial (modified) Mathieu function of the first kind 'Se' or its first derivative   or  
mathieu_rad_fey compute even radial (modified) Mathieu function of the second kind 'Fey' or its first derivative   or  
mathieu_rad_gey compute odd radial (modified) Mathieu function of the second kind 'Gey' or its first derivative   or  

During unit-testing all functions were tested against known tables: eigenvalues, expansion coefficients and angular functions and its first derivatives were compared with [3, 9], radial functions and their first derivatives were compared with [9-12].

For eigenvalues we have demo, named "Stability chart for eigenvalues of Mathieu`s equations" (for comparison with DLMF 28.17 [4]).

For modified solutions we have demos:

All functions have examples with plots, some of them for comparison with [3, 4].

Calculation modes of elliptical membrane

This toolbox allows elliptical membrane mode calculation. We have two functions for this purpose.
Function Purpose
mathieu_rootfinder rootfinder for radial Mathieu function or its first derivative (finds q values of given radial function type with known order m and radial argument ξ0, which satisfies the equation RMF_m(q,xi0) = 0 (Dirichlet boundary condition) or RMF_m'(q,xi0) = 0 (Neumann boundary condition).
mathieu_membrane_mode calculate elliptical membrane mode for known semi-axes, mode numbers, mode type and boundary condition (Soft/Dirichlet or Hard/Neumann).

During unit-testing these functions were tested against tables and plots from [2, 13-18].

For elliptical membrane we have demos:

Example

// plot Even Soft membrane mode with m=3, n=2
a = 0.05; // semi-major axis
b = 0.03; // semi-minor axis
m = 3; // function order (angular variations)
n = 2; // number of q root (radial variations)
mathieu_membrane_mode(a, b, m, n, 'Mc1', %t, 101, 101);

Dependencies

This module depends on the assert module.

Authors

R.Coisson and G. Vernizzi, Parma University

X. K. Yang

2011 - DIGITEO - Michael Baudin

N. O. Strelkov, NRU MPEI

Licence

This toolbox is distributed under the Gnu General Public License, Version 2.

Bibliography

1. 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).

2. N.W. McLachlan, Theory and Application of Mathieu Functions, Oxford Univ. Press, 1947.

3. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.

4. Chapter 28 Mathieu Functions and Hill's Equation. Digital Library of Mathematical Functions. NIST. (online at http://dlmf.nist.gov/28).

5. J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chavez-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).

6. J. J. Stamnes and B. Spjelkavik. New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders. Pure Appl. Opt. 4 251-62, 1995.

7. L. Chaos-Cador, E. Ley-Koo. Mathieu functions revisited: matrix evaluation and generating functions. Revista Mexicana de Fisica, Vol. 48, p.67-75, 2002.

8. Julio C. Gutierrez-Vega, "Formal analysis of the propagation of invariant optical fields in elliptic coordinates", Ph. D. Thesis, INAOE, México, 2000. (online at http://homepages.mty.itesm.mx/jgutierr/).

9. S. Zhang and J. Jin. Computation of Special Functions. New York, Wiley, 1996.

10. G. Blanch and D. S. Clemm. Tables relating to the radial Mathieu functions. Volume 1. Functions of the First Kind. ARL, US Air Force. 1963. (online at HathiTrust).

11. G. Blanch and D. S. Clemm. Tables relating to the radial Mathieu functions. Volume 2. Functions of the Second Kind. ARL, US Air Force. 1963. (online at HathiTrust).

12. E. T. Kirkpatrick. Tables of Values of the Modified Mathieu Functions. Mathematics of Computation, Vol. 14, No. 70 (Apr., 1960), pp. 118-129. (online at AMS).

13. Wilson, Howard B., and Robert W. Scharstein. "Computing elliptic membrane high frequencies by Mathieu and Galerkin methods." Journal of Engineering Mathematics 57.1 (2007): 41-55. (online at http://scharstein.eng.ua.edu/ENGI1589.pdf or http://dx.doi.org/10.1007/s10665-006-9070-1 )

14. Neves, Armando GM. "Eigenmodes and eigenfrequencies of vibrating elliptic membranes: a Klein oscillation theorem and numerical calculations." Comm. Pure Appl. Anal. 2009. (online at http://www.ma.utexas.edu/mp_arc/c/09/09-174.pdf )

15. Shibaoka, Yoshio, and Fusako Iida. "On the free oscillation of water in a lake of elliptic boundary." The Journal of the Oceanographical Society of Japan. 21.3 (1965): 103-108. (online at http://www.terrapub.co.jp/journals/JO/JOSJ/pdf/2103/21030103.pdf )

16. Hamidzadeh, Hamid R., and L. Moxey. "Analytical modal analysis of thin-film flat lenses." Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 219.1 (2005): 55-59.

17. Lee, W. M. "Natural mode analysis of an acoustic cavity with multiple elliptical boundaries by using the collocation multipole method." Journal of Sound and Vibration 330.20 (2011): 4915-4929.

18. Gutiérrez-Vega, J., S. Chávez-Cerda, and Ramón Rodríguez-Dagnino. "Free oscillations in an elliptic membrane." Revista Mexicana de Fisica 45.6 (1999): 613-622. (online at http://optica.mty.itesm.mx/pmog/Papers/P001.pdf )


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