Fredholm Integral Equation of the Second Kind
المؤلف:
Arfken, G
المصدر:
Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press,
الجزء والصفحة:
...
30-12-2018
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Fredholm Integral Equation of the Second Kind
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:
where
 |
(6)
|
Now multiply both sides of (◇) by 
and integrate over 
.
 |
(7)
|
By (◇), the first term is just 
. Now define
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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