## Quick Reference

Any equation for an unknown function *f*(*x*),

*a*≤*x*≤*b*

, involving integrals of the function. An equation of the form

*f*(*x*) = ∫^{x}_{a}*K*(*x*,*y*) *f*(*y*) d*y* + *g*(*x*)

is a **Volterra equation** of the second kind.

The analogous equation with constant limits

*f*(*x*) = ∫^{b}_{a}*K*(*x*,*y*) *f*(*y*) d*y* + *g*(*x*)

is a **Fredholm equation** of the second kind. If the required function only appears under the integral sign it is a Volterra or Fredholm equation of the first kind; these are more difficult to treat both theoretically and numerically. The Volterra equation can be regarded as a particular case of the Fredholm equation where

*K*(*x*,*y*) = 0 for *y* > *x*

Fredholm equations of the second kind occur commonly in boundary-value problems in mathematical physics. Numerical techniques proceed by replacing the integral with a rule for numerical integration, leading to a set of linear algebraic equations determining approximations to *f*(*x*) at a set of points in

*a*≤*x*≤*b*

.

*Subjects:*
Computing.