Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-09T06:01:29.692Z Has data issue: false hasContentIssue false

8.—Constructive Methods for Solving the Exterior Neumann Problem for the Reduced Wave Equation in a Spherically Symmetric Medium*

Published online by Cambridge University Press:  14 February 2012

David Colton
Affiliation:
Department of Mathematics, University of Strathclyde
Wolfgang Wendland
Affiliation:
Fachbereich Mathematik, Technische Hochschule, Darmstadt.

Synopsis

An integral operator is constructed which maps solutions of the reduced wave equation defined in exterior domains onto solutions of n u+λ2(l+B(r))u = 0 (*) defined in exterior domains, where B(r) is a continuously differentiable function of compact support. This operator is then used to construct a solution to the exterior Neumann problem for (*) satisfying the Sommerfeld radiation condition at infinity. Such problems arise in connection with the scattering of acoustic waves in a non-homogeneous medium, and this paper gives a method for solving these problems which is suitable for analytic and numerical approximations.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1976

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

REFERENCES

1Delves, L. M. and Walsh, J.. Numerical solution of integral equations (Oxford: Clarendon, 1974).Google Scholar
2Eichler, M.. On the differential equation uxx + uyy + N(x)u = 0. Trans. Amer. Math. Soc. 65 (1949), 259278.Google Scholar
3Gilbert, R. P.. The construction of solutions for boundary value problems by function theoretic methods. SIAMJ. Math. Anal. 1 (1970), 96114.CrossRefGoogle Scholar
4Jager, W.. Zur Theorie der Schwingungsgleichung mit variablen Koeffizienten in Außengebieten. Math. Z. 102 (1967), 6288.Google Scholar
5Jones, D. S.. Integral equations for the exterior acoustic problem. Quart. J. Mech. Appl. Math. 27 (1974), 129142.CrossRefGoogle Scholar
6Leis, R.. Zur Monotonie der Eigenwerte selbstadjungierter elliptischer Differentialgleichungen. Math. Z. 96 (1967), 2632.CrossRefGoogle Scholar
7Levitan, B.. Generalized translation operators and some of their applications (Jerusalem: Israel Program for Scientific Translations, 1964).Google Scholar
8Vekua, I. N.. New methods for solving elliptic equations (New York: Wiley, 1967).Google Scholar
9Werner, P.. Randwertprobleme der mathematischen Akustik. Arch. Rational Mech. Anal. 10 (1962), 2966.CrossRefGoogle Scholar