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Asymptotic Solutions of Integral Equations with Convolution Kernels

Published online by Cambridge University Press:  20 January 2009

V. Hutson
V. Hutson
Affiliation:
Department of Applied MathematicsThe UniversitySheffield, 10
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Summary

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The equations considered are Fredholm integral equations of the second kind with regular kernels, whose argument depends only on the difference of the variables. Approximate solutions are sought for a given finite range of the eigenvalues, and for large values of the range of integration. Certain special conditions are imposed on the general form of the Fourier transforms of the kernel. Then it is shown that approximate solutions may be obtained in terms of the solutions of the corresponding (singular) Wiener-Hopf equations. Approximations to the eigenvalues are also found. It is shown that the eigenfunctions are unique, and that except possibly near the end points of the range, the solutions are of trigonometric type with the zeros of successive solutions interlacing.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 14 , Issue 1 , June 1964 , pp. 5 - 19
Copyright
Copyright © Edinburgh Mathematical Society 1964

References

REFERENCE

(1) Carrier, G. F., 50 Jahre Grenzschichtforschung (Edited by Görtler, H. and Tollmien, W.) (Vieweg: Braunschweig, 1955), 13–20.CrossRefGoogle Scholar
(2) Case, K. M., Ann. Phys., 2 (1957), 384–405.CrossRefGoogle Scholar
(3) Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. 1 (Interscience: New York, 1953).Google Scholar
(4) Fock, V.C.R. Acad. Sci. U.R.S.S., 36 (1942), 133–136.Google Scholar
(5) Grenander, U. and Szegö, G., Toeplitz Forms and their Applications (Univversity of California Press, 1958).CrossRefGoogle Scholar
(6) Krein, M. G., Uspekhi Math. Nauk., 13 (1958), 8, 5 (83), 3–120.Google Scholar
(7) Latter, R., Quart. App. Math., 16 (1958), 21–31.CrossRefGoogle Scholar
(8) Noble, B., The Wiener-Hopf Technique (Pergammon Press, London, 1958).Google Scholar
(9) Paley, E. and Wiener, N.Fourier Transforms (Amer. Math. Soc. Colloquium Publication 19, New York, 1934).Google Scholar
(10) Titchmarsh, E. C., The Theory of Fourier Integrals (Clarendon Press, Oxford, 1948).Google Scholar
(11) Tricomi, F. G., Integral Equations (Interscience, New York, 1957).Google Scholar
(12) Widom, H., Trans. Amer. Math. Soc., 88 (1958), 491–522.CrossRefGoogle Scholar
(13) Widom, H., Trans. Amer. Math. Soc., 100 (1961), 252–262.CrossRefGoogle Scholar