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Date: 12-6-2018
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An integral equation of the form
(1) |
(2) |
The solution to a general Fredholm integral equation of the second kind is called an integral equation Neumann series.
A Fredholm integral equation of the second kind with separable integral kernel may be solved as follows:
(3) |
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(4) |
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(5) |
where
(6) |
Now multiply both sides of (◇) by and integrate over .
(7) |
By (◇), the first term is just . Now define
(8) |
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(9) |
so (◇) becomes
(10) |
Writing this in matrix form,
(11) |
so
(12) |
(13) |
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 865, 1985.
Baker, C. T. H. The Numerical Treatment of Integral Equations. Oxford, England: Clarendon Press, pp. 358-360, 1977.
Pearson, C. E. Handbook of Applied Mathematics. New York: Van Nostrand, 1990.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Fredholm Equations of the Second Kind." §18.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 782-785, 1992.
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