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# Fredholm Integral Equation of the Second Kind

 (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) (4) (5)

where

 (6)

Now multiply both sides of (◇) by and integrate over .

 (7)

By (◇), the first term is just . Now define

 (8) (9)

so (◇) becomes

 (10)

Writing this in matrix form,

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so

 (12)
 (13)

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