Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-12-01T03:48:46.662Z Has data issue: false hasContentIssue false
  • English
  • Français

Riemann's method and the characteristic value and cauchy problems for the damped wave equation†

Published online by Cambridge University Press:  18 May 2009

Eutiquio C. Young
Eutiquio C. Young
Affiliation:
Florida State UniversityTallahassee, Florida32306
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.

Riemann's method for solving the Cauchy problem for hyperbolic differential equations in two independent variables has been extended in a number of papers [4], [5], [2] to the wave equation in space of higher dimensions. The method, which consists in the determination of a so-called Riemann function, hinges on the solution of a characteristic value problem. Accordingly, if Riemann's method is to be used in solving a characteristic value problem, one will have to consider another characteristic value problem and thus the process becomes circular. This difficulty was first overcome by Protter [7] in solving the characteristic value problem for the wave equation in three variables. There he employed a variation of Riemann's method developed by Martin [5]. Martin's result was later extended by Diaz and Martin [2] to the wave equation in an arbitrary number of variables. This made it possible to extend Protter's result to the wave equation in space of higher dimensions [8].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 10 , Issue 2 , July 1969 , pp. 147 - 152
Copyright
Copyright © Glasgow Mathematical Journal Trust 1969

References

REFERENCES

1.d'Adhemar, R., Sur une équation aux derivées partielles du type hyperbolique, Rendiconti del Circolo Matematico di Palermo 20 (1905).Google Scholar
2.Diaz, J. B. and Martin, M. H., Riemann's method and the problem of Cauchy II, Proc. Amer. Math. Soc. 3 (1952), 476–483.Google Scholar
3.Garabedian, P. R., Partial differential equations (New York, 1964).Google Scholar
4.Lewy, H., Verallgemeinerung der Riemannschen methode auf mehr dimensionen, Nachr. Akad. Wiss. Gottingen (1928), 118–123.Google Scholar
5.Martin, M. H., Riemann's method and the problem of Cauchy, Bull. Amer. Math. Soc. 57 (1951), 238–249.CrossRefGoogle Scholar
6.Owens, O. G., Polynomial solutions of the cylindrical wave equation, Duke Math. J. 23 (1956), 371–383.CrossRefGoogle Scholar
7.Protter, M. H., The characteristic initial value problem for the wave equation and Riemann's method, Amer. Math. Monthly 61 (1954), 702–705.CrossRefGoogle Scholar
8.Young, E. C., The characteristic value problem for the wave equation in n dimensions, J. Math. Mech. 17 (1968), 885–889.Google Scholar
9.Young, E. C., On the Cauchy problem for the damped wave equation, J. Differential Equations 3 (1967), 228–235.CrossRefGoogle Scholar