An Initial-Value Theory for Fredholm Integral Equations With Semidegenerate Kernels

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An Initial-Value Theory for Fredholm Integral Equations With Semidegenerate Kernels

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Journal of the ACM (JACM) archive
Volume 17 , Issue 3 (July 1970) table of contents

Pages: 412 - 419

Year of Publication: 1970

ISSN:0004-5411

Authors

H. H. Kagiwada	Department of Electrical Engineering, University of Southern California, Los Angeles, California and The RAND Corporation, Santa Monica, California
R. Kalaba	Department of Electrical Engineering, University of Southern California, Los Angeles, California and The RAND Corporation, Santa Monica, California

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ACM New York, NY, USA

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Downloads (6 Weeks): 3, Downloads (12 Months): 38, Citation Count: 1

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ABSTRACT

The Fredholm integral equation where the kernel is semidegenerate has many applications. The solution of this integral equation may be studied as a function of the upper limit of integration x, while t remains fixed. It is shown that the solution satisfies an initial-value problem. This reformulation is well suited to numerical solution by analog and digital computers. The present paper is one of a series on initial-value methods for Fredholm integral equations. Its considerations are of practical significance since an arbitrary kernel may be approximated by a degenerate kernel to a desired degree of accuracy using standard techniques. Furthermore, the important cases in which the kernel is a Green's function and in which the integral equation is a Volterra equation are both covered by this treatment.

REFERENCES

Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.

	1	KA6IWADA, H. H., AND KALABA, R.E. An initial-value method for Fredholm integral' equations of convolution type. Int. J. Comp. Math. (to appear).
	2	-- AND An initial-value method suitable for the computation of certain Fredholm resolvents. J. Math. Phys. Sci. 1, 1 (1967), 109-122.
	3	-- AND --. Initial-value methods for the basic boundary value problem and integra~ equation of radiative transfer. J. Comp. Phys. I, 3 (1967), 322-329.
	4	, , AND SCHUMITZKY, A. An initial-value method for Fredholm integral equations. J. Math. Anal. Appl. 19, 1 (1967), 197-203.
	5	, , AND UENO, S. Invariant imbedding and Fredholm integral equations with Pincherle-Coursat kernels. RM-5599-PR, RAND Corp., Santa Moniea, Calif., April 1968.
	6	BELLMAN, R. E., KAGIWADA, H. H., AND KALABA, R. E. Numerical results for the auxiliary equation of radiative transfer. J. Quant. Spect. Radiat. Transfer 6, 3 (1966), 291-310.
	7	KAGIWADA, H. H., AND ~ALABA, R.E. A new initial-value method for internal intensities in radiative transfer. Astrophys. J. 147, 1 (1967), 301-309.
	8	BUELL, J., KAGIWADA, H. H., KALABA, R. E., McNABB, A., AND SCHUMITZKY, A. Computation of the resolvent for the auxiliary equation of radiative transfer. RM-5520-PR, RAND Corp., Santa Monica, Calif., Jan. 1968.
	9	KAGIWADA, H. H., AND KALABA, R. E. A practical method for determining Green's functions using Hadamard's variational formula. J. Optimization Theory Appl. 1, 1 (1967), 33-39. {1O} SCItMAEDEKE, W. Approximate solutions for Volterra integral equations of the first kind. J. Math. Anal. Appl. P3 (1968), 604-613.