Abstract
An equation for a function u (x 1, x 2, ..., x n ) of n independent variables x 1, x 2, ..., x n , in the simplest case for a function y(x), is called an integral equation when it involves an integral with the function u appearing in its integrand and with at least one of the arguments of u among its variables of integration. When the equation also involves somewhere a derivative of u, it is called an integro-differential equation.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Reference
The theorems for regular integral equations may easily be carried over to cases in which fewer assumptions are made about the kernel. The definitions of regular and singular integral equations used here follow those in Ph. Frank and R. V. Mises: Differential-and Integralgleichungen der Mechanik and Physik, 2nd ed., Vol. 1, p. 535. Brunswick 1930.
See, for instance, Ph. Frank and R. y. Misas: (see last footnote). — Courant, R., and D. Hilbert: Methods of mathematical physics, 1st English ed., Vol. I. New York: Interscience Publishers, Inc. 1953. — Hamel, G.: Integralgleichungen, 2nd ed. Berlin 1949. — Schmeidler, W.: Integralgleichungen mit Anwendungen in Physik und Technik, Vol. I, Lineare Integralgleichungen, 611 pp. Leipzig 1950.
See, for instance, W. Schmeidler: (see last footnote) pp. 328–360. — Collatz, L.: Eigenwertaufgaben, pp. 90–109. Leipzig 1949.
Gauss’s and Chebyshev’s quadrature formulae are recommended by E. J. NYSTRöM: ‘Ober die praktische Auflösung von linearen Integralgleichungen und Anwendungen auf Randwertaufgaben der Potentialtheorie. Commentationes physico-mathematicae. Acta Soc. Sci. Fenn. 4, Nr. 15, 1–52. Helsingfors 1928. Error estimates are given by L. V. kantorovich and V. I. krylov: Näherungsmethoden der höheren Analysis, pp. 94–155. Berlin 1956.
Nyström, E. J.: (see last footnote).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1960 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Collatz, L. (1960). Integral and functional equations. In: The Numerical Treatment of Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05500-7_6
Download citation
DOI: https://doi.org/10.1007/978-3-662-05500-7_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-05456-7
Online ISBN: 978-3-662-05500-7
eBook Packages: Springer Book Archive