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Collocation methods for the solution of eigenvalue problems for singular ordinary differential equations

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Języki publikacji
EN
Abstrakty
EN
We demonstrate that eigenvalue problems for ordinary differential equations can be recast in a formulation suitable for the solution by polynomial collocation. It is shown that the well-posedness of the two formulations is equivalent in the regular as well as in the singular case. Thus, a collocation code equipped with asymptotically correct error estimation and adaptive mesh selection can be successfully applied to compute the eigenvalues and eigenfunctions efficiently and with reliable control of the accuracy. Numerical examples illustrate this claim.
Rocznik
Strony
229--241
Opis fizyczny
Bibliogr. 19 poz., tab., rys., wykr.
Twórcy
autor
autor
autor
  • Vienna University of Technology. Institute for Analysis and Scientific Computing, Wiedner Hauptstrasse 8-10/101, A-1040 Wien, Austria, EU, w.auzinger@tuwien.ac.at
Bibliografia
  • [1] Ascher U., Christiansen J., Russell R. D., Collocation software for boundary value ODEs, ACM Transactions on Mathematical Software 7 (1981) 2, 209-222.
  • [2] Ascher U., Mattheij R. M. M., Russell R. D., Numerical solution of boundary value problems for ordinary differential equations, Prentice-Hall, Englewood Cliffs, NJ, 1988.
  • [3] Auzinger W., Kneisl G., Koch O., Weinmuller E., A collocation code for boundary value problems in ordinary differential equations, Numer. Algorithms 33 (2003), 27-39.
  • [4] Auzinger W., Koch O., Praetorius D., Weinmuller E., New a posteriori error estimates for singular boundary value problems, Numer. Algorithms 40 (2005), 79-100.
  • [5] Auzinger W., Koch O., Weinmuller E., Collocation methods for boundary value problems with an essential singularity, Large-Scale Scientific Computing (I. Lirkov, S. Margenov, J. Wasniewski, and P. Yalamov, eds.), Lecture Notes in Computer Science, vol. 2907, Springer Verlag, 2004, 347-354.
  • [6] ---, Analysis of a new error estimate for collocation methods applied to singular boundary value problems, SIAM J. Numer. Anal. 42 (2005) 6, 2366-2386.
  • [7] ---, Efficient mesh selection for collocation methods applied to singular BVPs, J. Comput. Appl. Math. 180 (2005), no. 1, 213-227.
  • [8] Bailey P.B., Everitt W.N., Weidmann J., Zettl A., Regular approximations of singular Sturm-Liouville problems, Results Math. 23 (1993), 3-22.
  • [9] Bailey P. B., Everitt W.N., Zettl A., Computing eigenvalues of singular Sturm-Liouville problems, Results Math. 20 (1991), 391-423.
  • [10] ---, Regular and singular Sturm-Liouville problems with coupled boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 505-514.
  • [11] de Hoog F.R., Weiss R., Difference methods for boundary value problems with a singularity of the first kind, SIAM J. Numer. Anal. 13 (1976), 775-813.
  • [12] ---, On the boundary value problem for systems of ordinary differential equations with a singularity of the second kind, SIAM J. Math. Anal. 11 (1980), 41-60.
  • [13] Keller H., Approximation methods for nonlinear problems with application to two-point boundary value problems, Math. Comp. 29 (1975), 464-474.
  • [14] Koch O., Asymptotically correct error estimation for collocation methods applied to singular boundary value problems, Numer. Math. 101 (2005), 143-164.
  • [15] Kreiss H.-O., Difference approximations for boundary and eigenvalue problems for ordinary differential equations, Math. Comp. 26 (1972) 119, 605-624.
  • [16] Naimark M. A., Linear Differential Operators, vol. II, Ungar, New York, 1968.
  • [17] Pryce J., Numerical solution of Sturm-Liouville problems, Oxford University Press, New York, 1993.
  • [18] Shampine L., Kierzenka J., A BVP solver based on residual control and the MATLAB PSE, ACM T. Math. Software 27 (2001), 299-315.
  • [19] Smiley M.W., Time-periodic solutions of nonlinear wave equations in balls, Oscillation, Bifurcation and Chaos, CMS Conference Proceedings, vol. 8, Canadian Mathematical Society, 1987, 287-297.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGH4-0007-0015
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