## Purpose

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.

We present both the Floquet solution and solution in angular and radial (modified) functions [2-4].

The toolbox has 14 demos.

All demos can be acessed from "? -> Scilab Demonstrations -> Mathieu functions"

## Dependencies

This module depends on the assert module.

## Authors

R.Coisson & G. Vernizzi, Parma University, 2001-2002.

X. K. Yang

N. O. Strelkov, NRU MPEI

Michael Baudin - DIGITEO - 2011

## Features

### Coordinate transformation

Almost all tasks, involving Mathieu functions have elliptical geometry. For coordinate transformation we have functions:

• `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

For better understanding of elliptical coordinates you can use:

• the first tab "Elliptic and Cartesian coordinates" of demo named "GUI for Mathieu Function Toolbox" (available for Scilab 5.5 and higher) (see Demos-0).

• the demo named "Elliptic and Cartesian coordinates" (see Demos-1).

### Floquet solution

For Floquet solution we have the following functions:

• `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:

• Plots ce_m(z,q), ce_m'(z,q), se_m(z,q), se_m'(z,q) for q = -25..25 and m = 0(1)..15 are available on the second tab "Angular Mathieu Functions" of demo named "GUI for Mathieu Function Toolbox" (available for Scilab 5.5 and higher) (see Demos-0).

• Plots of ce_m(z,q), se_m(z,q) of order 0-5 for q = 1,10 for comparison with Abramowitz and Stegun (see Demos-3, Figs. 20.2-20.5, [3, pp. 725-726]).

• Tables of ce_m(z,q) and ce'_m(z,q) for comparison with Abramowitz and Stegun (see Demos-4, Table 20.1 [3, p. 748]).

• Tables of se_m(z,q) and se'_m(z,q) for comparison with Abramowitz and Stegun (see Demos-5, Table 20.1 [3, p. 749]).

• Plots of ce_m(z,q), se_m(z,q) of order 0-3 for q = 0-30 and z = [0, π] (Fig. 2, [see Demos-6, 5, p. 235]).

### Solution in angular and radial (modified) functions

We present solutions of Mathieu equation as angular Mathieu functions [2,3]: ce_m(z,q) and se_m(z,q)  and solutions of modified Mathieu equation (2) as radial Mathieu functions [2, 3]: Mc_m(1)(z,q), Ce_m(z,q), Mc_m(2)(z,q), Gey_m(z,q) and Ms_m(1)(z,q), Se_m(z,q), Ms_m(2)(z,q), Fey_m(z,q).

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: * `mathieu_Arm` compute expansion coefficients 'Arm' and eigenvalue 'am' for even angular and radial Mathieu functions

• `mathieu_Brm` compute expansion coefficients 'Brm' and eigenvalue 'bm' for odd angular and radial Mathieu functions

• `mathieu_ang_ce` compute even angular Mathieu function 'ce' or its first derivative

• `mathieu_ang_se` compute odd angular Mathieu function 'se' or its first derivative

• `mathieu_rad_mc` compute even radial (modified) Mathieu function 'Mc' or its first derivative (kinds 1 and 2)

• `mathieu_rad_ms` compute odd radial (modified) Mathieu function 'Ms' or its first derivative (kinds 1 and 2)

• `mathieu_rad_ce` compute even radial (modified) Mathieu function of the first kind 'Ce' or its first derivative

• `mathieu_rad_se` compute odd radial (modified) Mathieu function of the first kind 'Se' or its first derivative

• `mathieu_rad_fey` compute even radial (modified) Mathieu function of the second kind 'Fey' or its first derivative

• `mathieu_rad_gey` compute odd radial (modified) Mathieu function of the second kind 'Gey' or its first derivative

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" (see Demos-2, for comparison with http://dlmf.nist.gov/28.17 ).

For modified solutions we have demos:

• Plots of Mc_m(1,2)(z,q), Mc_m(1,2)'(z,q); Ms_m(1,2)(z,q), Ms_m(1,2)'(z,q); Ce_m(z,q), Ce_m'(z,q); Se_m(z,q), Se_m'(z,q); Fey_m(z,q), Fey_m'(z,q); Gey_m(z,q), Gey_m'(z,q) for q = 1..11 and m = 0(1)..15 are available on the third tab "Radial Mathieu Functions" of demo named "GUI for Mathieu Function Toolbox" (available for Scilab 5.5 and higher) (see Demos-0).

• Plots of Mc_m(1)(z,q), Ms_m(1)(z,q) of order 0-2 for q = 0-3 and z = [0, π] (see Demos-7).

• Plots of Mc_m(2)(z,q), Ms_m(2)(z,q) of order 0-2 for q = 0-3 and z = [0, π] (see Demos-8).

• Plots of Ce_m(z,q), Se_m(z,q) of order 0-2 for q = 0-3 and z = [0, π] (see Demos-9).

• Plots of Fey_m(z,q), Gey_m(z,q) of order 0-2 for q = 0-3 and z = [0, π] (see Demos-10).

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:

• `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,`ξ0`) = 0 (Dirichlet boundary condition) or `RMF_m`'(q,`ξ0`) = 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:

• The fourth tab "Elliptic membrane" of demo named "GUI for Mathieu Function Toolbox" (available for Scilab 5.5 and higher) (see Demos-0).

• Elliptic membrane: comparison of 2 even soft modes (see Demos-11).

• Elliptic membrane: comparison of 4 even & odd, soft & hard (see Demos-12.

• Elliptic membrane: 3D surface plot of a even soft mode with m=3, n=3 (see Demos-13).

## 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);`

## 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. 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).
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. Gutiérrez-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,
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 http://catalog.hathitrust.org/Record/000585710).
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 http://catalog.hathitrust.org/Record/000585710).
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 http://www.ams.org/journals/mcom/1960-14-070/S0025-5718-1960-0113288-4/S0025-5718-1960-0113288-4.pdf).
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).