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)
|
See also
Fredholm Integral Equation of the First Kind,
Integral
Equation,
Integral Equation Neumann
Series,
Volterra Integral
Equation of the First Kind,
Volterra
Integral Equation of the Second Kind
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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.Referenced on Wolfram|Alpha
Fredholm Integral
Equation of the Second Kind
Cite this as:
Weisstein, Eric W. "Fredholm Integral Equation of the Second Kind." From MathWorld--A Wolfram Web Resource.
https://mathworld.wolfram.com/FredholmIntegralEquationoftheSecondKind.html
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