On numerical solution of multiparameter Sturm-Liouville spectral problems

T. Levitina

Banach Center Publications (1994)

  • Volume: 29, Issue: 1, page 275-281
  • ISSN: 0137-6934

Abstract

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The method proposed here has been devised for solution of the spectral problem for the Lamé wave equation (see [2]), but extended lately to more general problems. This method is based on the phase function concept or the Prüfer angle determined by the Prüfer transformation cotθ(x) = y'(x)/y(x), where y(x) is a solution of a second order self-adjoint o.d.e. The Prüfer angle θ(x) has some useful properties very often being referred to in theoretical research concerning both single- and multi-parameter Sturm-Liouville spectral problems (see e.g. [6,14,5]). All these properties may be useful for numerical solution of the above problems as well. For an account of numerical methods for solving the single-parameter Sturm-Liouville spectral problem by means of a modified Prüfer transformation one is referred to [1,11,9].

How to cite

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Levitina, T.. "On numerical solution of multiparameter Sturm-Liouville spectral problems." Banach Center Publications 29.1 (1994): 275-281. <http://eudml.org/doc/262571>.

@article{Levitina1994,
abstract = {The method proposed here has been devised for solution of the spectral problem for the Lamé wave equation (see [2]), but extended lately to more general problems. This method is based on the phase function concept or the Prüfer angle determined by the Prüfer transformation cotθ(x) = y'(x)/y(x), where y(x) is a solution of a second order self-adjoint o.d.e. The Prüfer angle θ(x) has some useful properties very often being referred to in theoretical research concerning both single- and multi-parameter Sturm-Liouville spectral problems (see e.g. [6,14,5]). All these properties may be useful for numerical solution of the above problems as well. For an account of numerical methods for solving the single-parameter Sturm-Liouville spectral problem by means of a modified Prüfer transformation one is referred to [1,11,9].},
author = {Levitina, T.},
journal = {Banach Center Publications},
keywords = {Lamé wave equation; Prüfer transformation; multi-parameter Sturm-Liouville spectral problems},
language = {eng},
number = {1},
pages = {275-281},
title = {On numerical solution of multiparameter Sturm-Liouville spectral problems},
url = {http://eudml.org/doc/262571},
volume = {29},
year = {1994},
}

TY - JOUR
AU - Levitina, T.
TI - On numerical solution of multiparameter Sturm-Liouville spectral problems
JO - Banach Center Publications
PY - 1994
VL - 29
IS - 1
SP - 275
EP - 281
AB - The method proposed here has been devised for solution of the spectral problem for the Lamé wave equation (see [2]), but extended lately to more general problems. This method is based on the phase function concept or the Prüfer angle determined by the Prüfer transformation cotθ(x) = y'(x)/y(x), where y(x) is a solution of a second order self-adjoint o.d.e. The Prüfer angle θ(x) has some useful properties very often being referred to in theoretical research concerning both single- and multi-parameter Sturm-Liouville spectral problems (see e.g. [6,14,5]). All these properties may be useful for numerical solution of the above problems as well. For an account of numerical methods for solving the single-parameter Sturm-Liouville spectral problem by means of a modified Prüfer transformation one is referred to [1,11,9].
LA - eng
KW - Lamé wave equation; Prüfer transformation; multi-parameter Sturm-Liouville spectral problems
UR - http://eudml.org/doc/262571
ER -

References

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  1. [1] A. A. Abramov, Methods of solution of some linear problems, doctoral dissertation, Computing Center Acad. Sci. USSR, Moscow 1974 (in Russian). 
  2. [2] A. A. Abramov, A. L. Dyshko, N. B. Konyukhova and T. V. Levitina, Computation of angular wave functions of Lamé by means of solution of auxiliary differential equations, Zh. Vychisl. Mat. i Mat. Fiz. 29 (6) (1989), 813-830 (in Russian); English transl.: USSR Comput. Math. and Math. Phys. 29 (1989). Zbl0693.65007
  3. [3] F. M. Arscott and B. D. Sleeman, High-frequency approximations to ellipsoidal wave functions, Mathematika 17 (1970), 39-46. Zbl0197.33901
  4. [4] F. M. Arscott, P. J. Taylor and R. V. M. Zahar, On the numerical construction of ellipsoidal wave functions, Math. Comp. 40 (1983), 367-380. Zbl0538.65007
  5. [5] P. A. Binding and P. J. Browne, Asymptotics of eigencurves for second order ordinary differential equations, part I, J. Differential Equations 88 (1990), 30-45, part II, ibid. 89 (1991), 224-243. Zbl0723.34073
  6. [6] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York 1955. Zbl0064.33002
  7. [7] M. Faerman, The completeness and expansion theorem associated with multiparameter eigenvalue problem in ordinary differential equations, J. Differential Equations 5 (1969), 197-213. Zbl0165.10003
  8. [8] M. V. Fedoryuk, Diffraction of waves by a tri-axial ellipsoid, Differentsial'nye Uravneniya 25 (11) (1989), 1990-1995 (in Russian). Zbl0699.35052
  9. [9] D. I. Kitoroagè, N. V. Konyukhova and B. S. Pariĭskiĭ, A Modified Phase Function Method for Problems Concerning Bound States of Particles, Soobshch. Prikl. Mat., Vychisl. Tsentr Akad. Nauk SSSR, Moscow 1986 (in Russian). 
  10. [10] T. V. Levitina, Conditions of applicability of an algorithm for solution of two-parameter self-adjoint boundary value problems, Zh. Vychisl. Mat. i Mat. Fiz. 31 (5) (1991), 689-697 (in Russian); English transl.: USSR Comput. Math. and Math. Phys. 31 (1991). Zbl0751.65054
  11. [11] T. V. Pak, A study of some singular problems with parameters for systems of ordinary differential equations and computation of spheroidal wave functions, thesis, Vychisl. Tsentr Akad. Nauk SSSR, 1986 (in Russian). 
  12. [12] R. G. D. Richardson, Theorems of oscillation for two linear differential equations of the second order with two parameters, Trans. Amer. Math. Soc. 13 (1912), 22-34. Zbl43.0400.03
  13. [13] B. D. Sleeman, Singular linear differential operators with many parameters, Proc. Roy. Soc. Edinburgh. Sect. A 71 (1973), 199-232. Zbl0323.34017
  14. [14] L. Turyn, Sturm-Liouville problems with several parameters, J. Differential Equations 38 (3) (1980), 239-259. Zbl0421.34023

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