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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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The inverse Sturm-Liouville problem and the Rayleigh-Ritz method
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by Ole H. Hald PDF
Math. Comp. 32 (1978), 687-705 Request permission

Abstract:

In this paper we present an algorithm for solving the inverse Sturm-Liouville problem with symmetric potential and Dirichlet boundary conditions. The algorithm is based on the Rayleigh-Ritz method for calculating the eigenvalues of a two point boundary value problem, and reduces the inverse problem for the differential equation to a nonstandard discrete inverse eigenvalue problem. It is proved that the solution of the discrete problem converges to the solution of the continuous problem. Finally, we establish the stability of the method and give numerical examples.
References
  • Lars-Erik Andersson, On the effective determination of the wave operator from given spectral data in the case of a difference equation corresponding to a Sturm-Liouville differential equation, J. Math. Anal. Appl. 29 (1970), 467–497. MR 267780, DOI 10.1016/0022-247X(70)90062-4
    • G. E. BACKUS & J. F. GILBERT, "Numerical applications of a formalism for geophysical inverse problems," Geophys. J. Roy. Astronom. Soc., v. 13, 1967, pp. 247-276.
  • Victor Barcilon, Iterative solution of the inverse Sturm-Liouville problem, J. Mathematical Phys. 15 (1974), 429–436. MR 342069, DOI 10.1063/1.1666664
  • Göran Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math. 78 (1946), 1–96 (German). MR 15185, DOI 10.1007/BF02421600
  • P. G. Ciarlet, M. H. Schultz, and R. S. Varga, Numerical methods of high-order accuracy for nonlinear boundary value problems. III. Eigenvalue problems, Numer. Math. 12 (1968), 120–133. MR 233517, DOI 10.1007/BF02173406
  • Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
  • R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • Shmuel Friedland, Inverse eigenvalue problems, Linear Algebra Appl. 17 (1977), no. 1, 15–51. MR 472861, DOI 10.1016/0024-3795(77)90039-8
  • F. R. Gantmacher and M. G. Kreĭn, Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme, Mathematische Lehrbücher und Monographien, I. Abteilung, Bd. V, Akademie-Verlag, Berlin, 1960 (German). Wissenschaftliche Bearbeitung der deutschen Ausgabe: Alfred Stöhr. MR 0114338
  • I. M. Gel′fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. (2) 1 (1955), 253–304. MR 0073805
  • K. P. Hadeler, Ein inverses Eigenwertproblem, Linear Algebra Appl. 1 (1968), no. 1, 83–101 (German, with English summary). MR 227189, DOI 10.1016/0024-3795(68)90051-7
    • O. H. HALD, On Discrete and Numerical Inverse Sturm-Liouville Problems, Ph.D. Thesis, New York Univ., New York, N. Y., 1972.
  • Ole H. Hald, The inverse Sturm-Liouville problem with symmetric potentials, Acta Math. 141 (1978), no. 3-4, 263–291. MR 505878, DOI 10.1007/BF02545749
    • G. H. HARDY, J. E. LITTLEWOOD & G. PÓLYA, Inequalities, 2nd ed., Cambridge Univ. Press, London, 1952.
  • Harry Hochstadt, The inverse Sturm-Liouville problem, Comm. Pure Appl. Math. 26 (1973), 715–729. Collection of articles dedicated to Wilhelm Magnus. MR 330607, DOI 10.1002/cpa.3160260514
  • Harry Hochstadt, Well-posed inverse spectral problems, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), no. 7, 2496–2497. MR 372313, DOI 10.1073/pnas.72.7.2496
  • Tosio Kato, On the upper and lower bounds of eigenvalues, J. Phys. Soc. Japan 4 (1949), 334–339. MR 38738, DOI 10.1143/JPSJ.4.334
  • M. G. Kreĭn, Solution of the inverse Sturm-Liouville problem, Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 21–24 (Russian). MR 0039895
  • M. G. Kreĭn, Determination of the density of a nonhomogeneous symmetric cord by its frequency spectrum, Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 345–348 (Russian). MR 0042045
  • Françoise Laborde, Sur un problème inverse d’un problème de valeurs propres, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A153–A156 (French). MR 241441
  • Norman Levinson, The inverse Sturm-Liouville problem, Mat. Tidsskr. B 1949 (1949), 25–30. MR 32067
  • B. M. Levitan, Regularized traces and conditions of smooth periodicity of the potential of a Sturm-Liouville equation, Sibirsk. Mat. Zh. 22 (1981), no. 2, 137–148, 237 (Russian). MR 610774
  • V. A. Marčenko, Concerning the theory of a differential operator of the second order, Doklady Akad. Nauk SSSR. (N.S.) 72 (1950), 457–460 (Russian). MR 0036916
  • Paul Morel, Des algorithmes pour le problème inverse des valeurs propres, Linear Algebra Appl. 13 (1976), no. 3, 251–273 (French). MR 395187, DOI 10.1016/0024-3795(76)90100-2
    • F. I. NIORDSON, "A method for solving inverse eigenvalue problems," in Recent Progress in Applied Mechanics, The Folke Odquist Volume (B. Broberg, J. Hult & F. Niordson, Editors), Almquist & Wiksell, Stockholm, 1967, pp. 375-382.
  • G. N. de Oliveira, Note on an inverse characteristic value problem, Numer. Math. 15 (1970), 345–347. MR 266415, DOI 10.1007/BF02165126
  • Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
    • Tables Relating to Mathieu Functions, National Bureau of Standards, Columbia Univ. Press, 1951.
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Math. Comp. 32 (1978), 687-705
  • MSC: Primary 65L15
  • DOI: https://doi.org/10.1090/S0025-5718-1978-0501963-2
  • MathSciNet review: 0501963