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4.—A Cauchy Problem for an Ordinary Integro-differential Equation*

Published online by Cambridge University Press:  14 February 2012

E. A. Catchpole
E. A. Catchpole
Affiliation:
Fluid Mechanics Research Institute, University of Essex†.
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Synopsis

In this paper we study an ordinary second-order integro-differential equation (IDE) on a finite closed interval. We demonstrate the equivalence of this equation to a certain integral equation, and deduce that the homogeneous IDE may have either 2 or 3 linearly independent solutions, depending on the value of a parameter λ. We study a Cauchy problem for the IDE, both by this integral equation approach and by an independent approach, based on the perturbation theory for linear operators. We give necessary and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides—these conditions again depend on λ—and specify the behaviour of the IDE when these conditions are not satisfied. At the end of the paper some examples are given of the type of behaviour described.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 72 , Issue 1 , 1974 , pp. 39 - 55
Copyright
Copyright © Royal Society of Edinburgh 1974

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References

References To Literature

Bykov, Ya. V., 1961. On some Constructive Methods of Solution of Integral and Integro-differential Equations. Frunze. (Russian.)Google Scholar
Catchpole, E. A., 1971. An integro-differential equation. Ph.D. thesis, University of Dundee.Google Scholar
Catchpole, E. A., 1973. An integro-differential operator. J. Lond. Math. Soc., 6, 513–523.CrossRefGoogle Scholar
Coddington, E. A. and Levinson, N., 1955. Theory of Ordinary Differential Equations. New York: McGraw-Hill.Google Scholar
Copson, E. T., 1935. An Introduction to the Theory of Functions of a Complex Variable. O.U.P.Google Scholar
Kato, T., 1966. Perturbation Theory for Linear Operators. Berlin: Springer-Verlag.Google Scholar
Lando, Yu. K., 1962. Adjoint Cauchy problems for a class of integro-differential equations. Dokl. Akad. Nauk. B.S.S.R., 6, 616–619.Google Scholar
Markushevich, A. I., 1965. Theory of Functions of a Complex Variable, II. New Jersey: Prentice- Hall.Google Scholar
Naimark, M. A., 1968. Linear Differential Operators, II. New York: Ungar.Google Scholar
Stakgold, I., 1967. Boundary Value Problems of Mathematical Physics, I. New York: Macmillan.CrossRefGoogle Scholar
Stone, M. H., 1932. Linear Transformations in Hilbert Space. New York: Am. Math. Soc.Google Scholar