Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T17:10:55.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of regular approximations to the spectra of singular fourth-order Sturm–Liouville problems

Published online by Cambridge University Press:  14 November 2011

Malcolm Brown ,
Leon Greenberg  and
Marco Marletta
Malcolm Brown
Affiliation:
Department of Computer Science, University of Wales—Cardiff, P.O. Box 916, Cardiff CF2 3XF, U.K.
Leon Greenberg
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland MD 20742, U.S.A.
Marco Marletta
Affiliation:
Department of Mathematics, University of Leicester, University Road, Leicester LEI 7RH, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove some new results which justify the use of interval truncation as a means of regularising a singular fourth-order Sturm–Liouville problem near a singular endpoint. Of particular interest are the results in the so-called lim-3 case, which has no analogue in second-order singular problems.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 5 , 1998 , pp. 907 - 944
Copyright
Copyright © Royal Society of Edinburgh 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Akhiezer, N. I. and Glazman, I. M.. Theory of Linear Operators in Hilbert Space, Vol. II (London: Pitman, 1981).Google Scholar
2Bailey, P. B., Everitt, W. N., Weidmann, J. and Zettl, A.. Regular approximation of singular Sturm–Liouville problems. Results Math. 23 (1993), 3–22.CrossRefGoogle Scholar
3Bailey, P. B., Everitt, W. N. and Zettl, A.. Computing eigenvalues of singular Sturm–Liouville problems. Results Math. 20 (1991), 391–423.CrossRefGoogle Scholar
4Bailey, P. B., Gordon, M. K. and Shampine, L. F.. Automatic solution of Sturm–Liouville problems. ACM Trans. Math. Software 4 (1978), 193–208.CrossRefGoogle Scholar
5Bennewitz, C., Brown, B. M., Evans, W. D., McCormack, D. K. R. and Marletta, M.. Computation of the M matrix for fourth order problems. Proc. Roy. Soc. London Sect. A 452 (1996), 1765–1788.Google Scholar
6Berkowitz, J.. On the discreteness of spectra of singular Sturm–Liouville problems. Comm. Pure Appl. Math. 12 (1959), 523–42.CrossRefGoogle Scholar
7Dunford, D. and Schwartz, J. T.. Linear Operators; Part II: Spectral Theory (New York: Interscience, 1963).Google Scholar
8Everitt, W. N. and Zettl, A.. Generalized symmetric ordinary differential expressions. I. The basic theory. Nieuw Arch. Wisk. (3) 27 (1979), 363–97.Google Scholar
9Fulton, C. T. and Pruess, S.. Mathematical software for Sturm–Liouville problems. ACM Trans. Math. Software 19 (1993), 360–76.Google Scholar
10Greenberg, L.. A Prüfer method for calculating eigenvalues of self-adjoint systems of ordinary differential equations, Parts 1 and 2 (University of Maryland Technical Report TR91-24, 1991).Google Scholar
11Greenberg, L. and Marletta, M.. Oscillation theory and numerical solution of fourth order Sturm–Liouville problems. IMA J. Numer. Anal. 15 (1995), 319–56.CrossRefGoogle Scholar
12Greenberg, L. and Marletta, M.. The code SLEUTH for solving fourth order Sturm–Liouville problems (submitted).Google Scholar
13Hutson, W. and Pym, J. S.. Applications of Functional Analysis and Operator Theory (Londond: Academic Press, 1980).Google Scholar
14Marletta, M.. Numerical solution of eigenvalue problems for hamiltonian systems. Adv. Comput. Math. 2 (1994), 155–84.CrossRefGoogle Scholar
15Naimark, M. A.. Linear Differential Operators; Part II: Linear Differentil Operators in Hilbert Space (New York: Frederick Ungar, 1968).Google Scholar
16Reed, M. and Simon, B.. Methods of Modern Mathematical Physics; I: Functional Analysis (London: Academic Press, 1972).Google Scholar
17Zettl, A.. Formally self-adjoint quasi-differential operators. Rocky Mountain J. Math. 5 (1975), 453–74.CrossRefGoogle Scholar