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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).

Calling Sequence

q_found_out = mathieu_rootfinder(m, xi0, q_start, q_delta, q_roots_tot, q_fun_tol, q_maxiter, func_name, fun_or_der, do_print, do_plot)

Parameters

m

the order of Mathieu function

xi0

the value of radial argument in elliptic coordinates ()

q_start

initial guess for q

q_delta

step increment on search of q

q_roots_tot

number of roots to be found

q_fun_tol

relative tolerance of zero value (fsolve tol)

q_maxiter

maximum number if iterations

func_name

radial Mathieu function type (should be on of 'Ce', 'Se', 'Fey', 'Gey', 'Mc1', 'Ms1', 'Mc2', 'Ms2')

fun_or_der

search roots of function (%t - Dirichlet boundary condition) or derivative (%f - Neumann boundary condition)

do_print

print debug infromation in console? (boolean)

do_plot

plot graphics of rootfinding? (boolean)

q_found_out

found values of q

Description

Mathieu_rootfinder finds q values of given radial Mathieu function (RMF) type with known order m and radial argument , which satisfies the equation (Dirichlet boundary condition) or (Neumann boundary condition).

Rootfinder uses bisection method and fsolve function.

During unit-testing rootfinder was tested by comparison with results from articles and books: [1, Table 5(a)], [2, Table 1]; [3, p. 299], [3, p. 302], [3, p. 304], [3, p. 308, Table 24]; [4, p. 105, Table 1]; [5, p. 58], [6, Fig. 9]; [7, pp. 618-619].

The results are very close.

Examples

// Comparison with [4, p. 105, Table 1]
  // 
  // Finding roots of derivative Ce0`(xi,q) = 0
  // 
  // a/b = 5/4  e = 0.600  q = 1.785
  // a/b = 2    e = 0.866  q = 8.57
  // a/b = 3    e = 0.943  q = 21.43
  //  
  printf("\n\nComparison with Y. Shibaoka and F. Iida [4, p. 105, Table 1], Ce0`(xi,q)=0:\n");
  func_name = "Ce";
  m = 0;
  fun_or_der = 0;

  rho_article = [5/4,2,3];
  q_article = [1.785,8.57,21.43];

  q_start = 0.1;
  q_delta = 0.05;
  q_roots_tot = 1;
  q_fun_tol = 1e-16;
  q_maxiter = 1000;

  do_print = 1;
  do_plot = 1;

  for i = 0:length(rho_article)-1
    rho = rho_article(i+1);
  
    e = sqrt(1-rho^(-2));
    xi0 = acosh(1/e);
  
    printf("\n   xi0 = %f \n",xi0);
  
    qs = mathieu_rootfinder(m,xi0,q_start,q_delta,q_roots_tot,q_fun_tol,q_maxiter,func_name,fun_or_der,do_print,do_plot);
  
    printf("\tq%1d1 (article) = %f \n",0,q_article(i+1));
    printf("\tq%1d1  (Scilab) = %f \n",0,qs(1));
    delta(i+1) = (100*(q_article(i+1) - qs(1)))/q_article(i+1);
    printf("\tdelta = %f %% \n", delta(i+1));
  end

See Also

Authors

N. O. Strelkov, NRU MPEI

Bibliography

1. 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 )

2. 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 )

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

4. 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 )

5. 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.

6. 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.

7. 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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