Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-10T11:10:03.931Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Elementary Converse Problems in Ordinary Differential Equations*

Published online by Cambridge University Press:  20 November 2018

D. E. Seminar
D. E. Seminar*
Affiliation:
Aarhus University, Denmark, York University, Toronto
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In studying differential equations, the usual task is to determine properties of the solutions of such equations from a knowledge of the coefficient functions. The converse question, namely, of determining the coefficient functions from properties of solutions, also has significance. It has been studied especially in the case of Sturm-Liouville equations.

A discussion of the inverse Sturm-Liouville problem can be found in [8, Chapter 8], where references are given to the work of W.A. Ambarzumiam, G. Borg, I.M. Gelfand, M.G. Krein, B.M. Levitan, N. Levinson and W.A. Marchenko on this problem. Work of a quite different character, but dealing also with questions of a converse type arising from Sturm-Liouville equations, has been done by O. Boruvka and his colleagues and students [2].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 5 , December 1968 , pp. 703 - 716
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

*

The results arose from discussions in a seminar on differential equations at Aarhus University, Denmark, during 1964-65. The participants were JɈtte Bretlau, Villy K. Christensen, Jens Jørgen Hoist, Margrethe Jørgensen, Tove Lund Jørgensen, Karen Skov Larsen, Lee Lorch, Niels Wendell Pedersen, Per Amdal Steffensen, Leif Hautop Sørensen, and Preben Dahl Vestergaard.

References

1. Appell, P., Sur la transformation des équations différentielles linéaires. C. R. Acad. Sci. Paris 91, (1880) 211-214.Google Scholar
2. Borůvka, O, Lineare differentialtransformationen 2. ordnung. (Akad. Verlag Berlin, 1967).Google Scholar
3. Burkill, J. C., The theory of ordinary differential equations. (Edinburgh and London, second edition, 1962).Google Scholar
4. Hartman, P., On differential equations and the function . Amer. Jour, of Math. 83, (1961) 154-188.Google Scholar
5. Landau, E., Differential and integral calculus. English translation (Chelsea, New York, 1951).Google Scholar
6. Lorch, L. and Szego, P., Higher monotonicity properties of certain Sturm-Liouville functions Acta Math. 109, (1963) 55-83.Google Scholar
7. Lorch, L. and Szego, P., Higher monotonicity properties of certain Sturm-Liouville functions. II. Bull. Acad. Polonaise Sci. Sér. Sci. Math. Astr. Phys. 11, (1963) 455-457.Google Scholar
8. Naimark, M.A. (Neumark), Linear differential operators, Part II. (New York, 1968; English translation (Revised), of Russian original, Moscow, 1954). [A German translation was published in Berlin in I960.]Google Scholar
9. Whittaker, E. T. and Watson, G.N., A course of modern analysis. (Cambridge, fourth edition,1927).Google Scholar