If the limits are fixed, an integral equation is called a Fredholm integral equation. If one limit is variable, it is
called a Volterra integral equation. If the unknown function is only under the integral sign, the equation is said to be of
the ``first kind.'' If the function is both inside and outside, the equation is called of the ``second kind.'' A Fredholm
equation of the first kind is of the form

(1) |

(2) |

(3) |

(4) |

A Kernel is separable if

(5) |

(6) |

where

(7) |

(8) |

(9) | |||

(10) |

so (8) becomes

(11) |

(12) |

(13) |

(14) |

**References**

Corduneanu, C. *Integral Equations and Applications.* Cambridge, England: Cambridge University Press, 1991.

Davis, H. T. *Introduction to Nonlinear Differential and Integral Equations.* New York: Dover, 1962.

Kondo, J. *Integral Equations.* Oxford, England: Clarendon Press, 1992.

Lovitt, W. V. *Linear Integral Equations.* New York: Dover, 1950.

Mikhlin, S. G.
*Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, 2nd rev. ed.*
New York: Macmillan, 1964.

Mikhlin, S. G. *Linear Integral Equations.* New York: Gordon & Breach, 1961.

Pipkin, A. C. *A Course on Integral Equations.* New York: Springer-Verlag, 1991.

Porter, D. and Stirling, D. S. G. *Integral Equations: A Practical Treatment, from Spectral Theory to Applications.*
Cambridge, England: Cambridge University Press, 1990.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Integral Equations and Inverse Theory.'' Ch. 18
in *Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.* Cambridge, England:
Cambridge University Press, pp. 779-817, 1992.

Tricomi, F. G. *Integral Equations.* New York: Dover, 1957.

© 1996-9

1999-05-26