The Fredholm

equation is an

integral equation of the form:

/b
g(t) = | K(s,t) f(s) ds
/a

where `g(t)`

is often called the "data" and `K(s,t)`

is the kernel of the equation. Fredholm equations are linear transformations of a function `f(t)`

, and as such, they can be seen as a convolution of `f(t)`

with a varying impulse response. Of particular interest is the case where `f(t)`

is unknown; one then can find `f(t)`

from `g(t)`

and `K(s,t)`

, if the kernel is invertible (that is, non singular).

If the upper limit of the integral is replaced by variable `t`

, i.e.:

/t
g(t) = | K(s,t) f(s) ds ,
/a

we get a Volterra equation. In this case, it is always possible to retrieve `f(t)`

from `g(t)`

since the discontinuity in the integration range breaks any smoothness that the kernel may have at `s = t`

. This is analogous to the case of a matrix equation involving a lower triangular matrix.

Fredholm and Volterra equations of this form are said to be "of the first kind"; the second kind being

/b
f(t) = k | K(s,t) f(s) ds + g(t)
/a

for the Fredholm equation, and of course

/t
f(t) = k | K(s,t) f(s) ds + g(t)
/a

for the Volterra equation, where `k`

is a given constant. Those last equations are analoguous to matrix eigenvalues problems.

A nonlinear Fredholm equation would be of the general form:

/t
g(t) = | f(s) K(s,t,f(s)) ds
/a