Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Asymptotic formulae for the eigenvalues of a two-parameter ordinary differential equation of the second order


Author: M. Faierman
Journal: Trans. Amer. Math. Soc. 168 (1972), 1-52
MSC: Primary 34B25
DOI: https://doi.org/10.1090/S0002-9947-1972-0296390-0
MathSciNet review: 0296390
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a two-point boundary value problem associated with an ordinary differential equation defined over the unit interval and containing the two parameters $ \lambda $ and $ \mu $. If for each real $ \mu $ we denote the $ m$th eigenvalue of our system by $ {\lambda _m}(\mu )$, then it is known that $ {\lambda _m}(\mu )$ is real analytic in $ - \infty < \mu < \infty $. In this paper we concern ourselves with the asymptotic development of $ {\lambda _m}(\mu )$ as $ \mu \to \infty $, and indeed obtain such a development to an accuracy determined by the coefficients of our differential equation. With suitable conditions on the coefficients of our differential equation, the asymptotic formula for $ {\lambda _m}(\mu )$ may be further developed using the methods of this paper. These results may be modified so as to apply to $ {\lambda _m}(\mu )$ as $ \mu \to - \infty $ if the coefficients of our differential equation are also suitably modified.


References [Enhancements On Off] (What's this?)

  • [1] R. G. D. Richardson, Theorems of oscillation for two linear differential equations of the second order, Trans. Amer. Math. Soc. 13 (1912), 22-34. MR 1500902
  • [2] -, Über die notwendig und hinreichenden Bedingungen für das bestehen eines Kleinschen Oszillationstheorems, Math. Ann. 73 (1912/13), 289-304. MR 1511734
  • [3] M. Faierman, Boundary value problems in differential equations, Ph.D. Dissertation, University of Toronto, June 1966.
  • [4] J. Meixner and F. W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen mit Anwendung auf physikalische und technische Problems, Die Grundlehren der math. Wissenschaften, Band 71, Springer-Verlag, Berlin, 1954. MR 16, 586. MR 0066500 (16:586g)
  • [5] M. J. O. Strutt, Reelle Eigenwerte verallgemeinerter Hillscher Eigenwertaufgaben 2. Ordnung, Math. Z. 49 (1944), 593-643. MR 6, 174. MR 0011508 (6:174b)
  • [6] R. E. Langer, The asymptotic solutions of certain linear ordinary differential equations of the second order, Trans. Amer. Math. Soc. 36 (1934), 90-106. MR 1501736
  • [7] -, The asymptotic solutions of ordinary linear differential equations of the second order with special reference to the Stokes phenomenon, Bull. Amer. Math. Soc. 40 (1934), 545-582. MR 1562910
  • [8] -, The asymptotic solutions of a linear differential equation of the second order with two turning points, Trans. Amer. Math. Soc. 90 (1959), 113-142. MR 21 #4270. MR 0105530 (21:4270)
  • [9] N. D. Kazarinoff, Asymptotic theory of second order differential equations with two simple turning points, Arch. Rational Mech. Anal. 2 (1958), 129-150. MR 20 #5919. MR 0099480 (20:5919)
  • [10] A. A. Dorodnicyn, Asymptotic laws of distribution of the characteristic values for certain special forms of differential equations of the second order, Uspehi Mat. Nauk 7 (1952), no. 6 (52), 3-96; English transl., Amer. Math. Soc. Transl. (2) 16 (1960), 1-101. MR 14, 876; MR 22 #8161. MR 0054137 (14:876a)
  • [11] R. W. McKelvey, The solutions of second order linear ordinary differential equations about a turning point of order two, Trans. Amer. Math. Soc. 79 (1955), 103-123. MR 16, 1023. MR 0069344 (16:1023f)
  • [12] A. Erdelyi et al., Higher transcendental functions. Vol. III, McGraw-Hill, New York, 1955. MR 16, 586. MR 0066496 (16:586c)
  • [13] E. T. Whittaker and G. N. Watson, Modern analysis, Cambridge Univ. Press, Cambridge, 1965.
  • [14] A. Erdelyi et al., Higher transcendental functions. Vol. II, McGraw-Hill, New York, 1953. MR 15, 419.
  • [15] M. Faierman, On a perturbation in a two-parameter ordinary differential equation of the second order, Canad. Math. Bull. 14 (1971), 25-33. MR 0291583 (45:674)
  • [16] -, Some properties of equations in integers, Canad. Math. Bull. (to appear). MR 0311982 (47:544)
  • [17] E. C. Titchmarsh, Eigenfunction expansions associated with second order differential equations. Part I, 2nd ed., Clarendon Press, Oxford, 1962. MR 31 #426. MR 0176151 (31:426)
  • [18] W. Wasow, Asymptotic expansions for ordinary differential equations, Pure and Appl. Math., vol. 14, Interscience, New York, 1965. MR 34 #3041. MR 0203188 (34:3041)
  • [19] E. C. Titchmarsh, Eigenfunction expansions associated with second order differential equations. Part II, Clarendon Press, Oxford, 1958. MR 20 #1065. MR 0094551 (20:1065)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34B25

Retrieve articles in all journals with MSC: 34B25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0296390-0
Keywords: Linear system, two parameters, real-valued functions, continuous functions, eigenvalues, absolute maximum, transition points, asymptotic integration, Weber equation, parabolic cylinder function, modified Bessel equation, modified Bessel function, matching of solutions, adjacent subintervals, main equation, inverse function theorem, perturbed equation, eigenfunctions, orthogonal properties of the eigenfunctions, equations in integers
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society