Fredholm equation

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 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

and the problem is, given the continuous kernel function K(t,s) and the function g(t), to find the function f(s).

If the kernel is a function only of the difference of its arguments, namely 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 K and f and therefore the solution is given by

where F_t and F_ω⁻¹ are the direct and inverse Fourier transforms, respectively.

An inhomogeneous Fredholm equation of the second kind is given as

${\displaystyle \varphi (t)=f(t)+\lambda \int _{a}^{b}K(t,s)\varphi (s)\,\mathrm {d} s.}$

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