| 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
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