In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to solve such type of equations, the Adomian decomposition method, is due to George Adomian.

A Fredholm equation is an integral equation in which the term containing the kernel function (defined below) has constants as integration limits. A closely related form is the Volterra integral equation which has variable integral limits.

An inhomogeneous Fredholm equation of the first kind is written as

${\displaystyle g(t)=\int _{a}^{b}K(t,s)f(s)\,\mathrm {d} s~,}$

and the problem is, given the continuous kernel function ${\displaystyle K}$ and the function ${\displaystyle g}$, to find the function ${\displaystyle f}$.

An important case of these types of equation is the case when the kernel is a function only of the difference of its arguments, namely ${\displaystyle K(t,s)=K(t-s)}$, and the limits of integration are ±∞, then the right hand side of the equation can be rewritten as a convolution of the functions ${\displaystyle K}$ and ${\displaystyle f}$ and therefore, formally, the solution is given by

where ${\displaystyle {\mathcal {F}}_{t}}$ and ${\displaystyle {\mathcal {F}}_{\omega }^{-1}}$ are the direct and inverse Fourier transforms, respectively. This case would not be typically included under the umbrella of Fredholm integral equations, a name that is usually reserved for when the integral operator defines a compact operator (convolution operators on non-compact groups are non-compact, since, in general, the spectrum of the operator of convolution with ${\displaystyle K}$ contains the range of ${\displaystyle {\mathcal {F}}{K}}$, which is usually a non-countable set, whereas compact operators have discrete countable spectra).

An inhomogeneous Fredholm equation of the second kind is given as