Integral equation

Summary

An integral equation is an equation in which the unknown function appears under an integral sign. The typical linear example is $\varphi(x) = f(x) + \lambda \int_a^b K(x, s) \varphi(s) \, ds$, where $f$ and the kernel $K$ are given and $\varphi$ is unknown. Many problems for differential equations, such as initial value problems and boundary value problems, can be rewritten as integral equations. The integral form often needs less smoothness and suits fixed point and operator methods.