Fredholm equation of the first type

Definition

A Fredholm equation of the first type is a particular type of integral equation that happens to be a linear integral equation. It looks like this:

$f(x)=\int _{a}^{b}K(x,t)\varphi (t)\,dt$

where $f$ and $K$ are known functions and $\varphi$ is the unknown function that we need to solve for. A functional solution to this is a functional solution such that the above holds for all values of $x$.

Related notions

• Fredholm equation of the second type refers to a situation where $\varphi$ appears both inside and outside the integral.
• Volterra equation of the first type
• Volterra equation of the second type