Abstract
Many important Fredholm integral equations have separable kernels which are finite-rank modifications of Volterra kernels. This class includes Green’s functions for Sturm—Liouville and other two-point boundary-value problems for linear ordinary differential operators. It is shown how to construct the Fredholm determinant, resolvent kernel, and eigenfunctions of kernels of this class by solving related Volterra integral equations and finite, linear algebraic systems. Applications to boundary-value problems are discussed, and explicit formulas are given for a simple example. Analytic and numerical approximation procedures for more general problems are indicated.
This research was sponsored by the United States Army under Contract No. DAA29-75-C0024.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Lovitt, W. V., Linear Integral Equations, Dover Publications, New York, New York, 1950.
Friedman, B., Principles and Techniques of Applied Mathematics, John Wiley and Sons, New York, New York, 1956.
Hochstadt, H., Integral Equations, John Wiley and Sons, New York, New York, 1973.
Widom, H., Lectures on Integral Equations, Van Nostrand—Reinhold Company, New York, New York, 1969.
Lalesco, T., Introduction a la Theorie des Equations Integrales, A. Herman et Fils, Paris, France, 1912.
Hellinger, E., and Toeplitz, O., Integralgleichungen und Gleichungen mit unendlichvielen Unbekannten, Chelsea Publishing Company, New York, New York, 1953.
Fredholivt, I., Sur une Classe d’Equations Fonctionnelles, Acta Mathematics, Vol. 27, pp. 152–156, 1955.
Forsyth, G. E., and Straus, L. W., The Souriau—Frame Characteristic Equation Algorithm on a Digital Computer, Journal of Mathematics and Physics, Vol. 34, pp. 152–156, 1955.
Drukarev, G. F., The Theory of Collisions of Electrons with Atoms, Soviet Physics JEPT, Vol. 4, pp. 309–320, 1957.
Brysk, H., Determinantal Solution of the Fredholm Equation with Green’s Function Kernel, Journal of Mathematical Physics, Vol. 4, pp. 1536–1538, 1963.
Aalto, S. K., Reduction of Fredholm Integral Equations with Green’s Function Kernels to Volterra Equations, Oregon State University, MS Thesis, 1966.
Manning, I., Theorem on the Determinantal Solution of the Fredholm Equation, Journal of Mathematical Physics, Vol. 5, pp. 1223–1225, 1964.
Yosida, K., Lectures on Differential and Integral Equations, John Wiley and Sons (Interscience Publishers), New York, New York, 1960.
Keller, H. B., Numerical Methods for Two-Point Boundary Value Problems, Blaisdell Publishing Company, Waltham, Massachusetts, 1968.
Householder, A. S., The Theory of Matrices in Numerical Analysis, Blaisdell Division of Ginn and Company, New York, New York, 1964.
Rall, L. B., Numerical Inversion of Green’s Matrices, University of Wisconsin, Madison, Wisconsin, Mathematics Research Center, Technical Summary Report No. 1149, 1971.
Rall, L. B., Numerical Inversion of a Class of Large Matrices, Proceedings of the 1975 Army Numerical Analysis and Computers Conference, US Army Research Office, Research Triangle Park, North Carolina, 1975.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1979 Springer Science+Business Media New York
About this chapter
Cite this chapter
Rall, L.B. (1979). Resolvent Kernels of Green’s Function Kernels and Other Finite-Rank Modifications of Fredholm and Volterra Kernels. In: Golberg, M.A. (eds) Solution Methods for Integral Equations. Mathematical Concepts and Methods in Science and Engineering, vol 18. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-1466-1_11
Download citation
DOI: https://doi.org/10.1007/978-1-4757-1466-1_11
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-1468-5
Online ISBN: 978-1-4757-1466-1
eBook Packages: Springer Book Archive