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1994 | 29 | 1 | 275-281
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On numerical solution of multiparameter Sturm-Liouville spectral problems

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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].
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  • Computing Center, Russian Academy of Sciences, Vavilova 40, 117967 Moscow GSP-1, Russia
  • [1] A. A. Abramov, Methods of solution of some linear problems, doctoral dissertation, Computing Center Acad. Sci. USSR, Moscow 1974 (in Russian).
  • [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).
  • [3] F. M. Arscott and B. D. Sleeman, High-frequency approximations to ellipsoidal wave functions, Mathematika 17 (1970), 39-46.
  • [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.
  • [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.
  • [6] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York 1955.
  • [7] M. Faerman, The completeness and expansion theorem associated with multiparameter eigenvalue problem in ordinary differential equations, J. Differential Equations 5 (1969), 197-213.
  • [8] M. V. Fedoryuk, Diffraction of waves by a tri-axial ellipsoid, Differentsial'nye Uravneniya 25 (11) (1989), 1990-1995 (in Russian).
  • [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] 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).
  • [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] 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.
  • [13] B. D. Sleeman, Singular linear differential operators with many parameters, Proc. Roy. Soc. Edinburgh. Sect. A 71 (1973), 199-232.
  • [14] L. Turyn, Sturm-Liouville problems with several parameters, J. Differential Equations 38 (3) (1980), 239-259.
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