# integral equation

**integral equation** Any equation for an unknown function *f*(*x*), *a*←*x*←*b*, involving integrals of the function. An equation of the form

is a *Volterra equation* of the second kind. The analogous equation with constant limits

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*.

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**integral equation**