---
tag: 8KJCN49V64
kind: concept
name: Integral equation of the second kind
msc: [45A05, 45B05]
specializes: [AYZCKG34QK]
sources:
  - url: https://en.wikipedia.org/wiki/Integral_equation
---

## Summary

An integral equation of the second kind is a [[linear integral equation|AYZCKG34QK]] of the form $\varphi(x) = f(x) + \lambda \int K(x, s) \varphi(s) \, ds$, where the unknown also appears outside the integral.
In operator form it reads $(I - \lambda K) \varphi = f$.
For reasonable kernels this equation is well behaved: Fredholm theory shows that it is uniquely solvable for all $f$ unless $\lambda$ is a characteristic value.
For small $\abs{\lambda}$ the solution is given by the [[Liouville–Neumann series|H1HWSWME6B]].

## Explanations

### L1 human

The unknown equals a known function plus a correction that depends on the unknown itself.
When the correction is small, one can solve the equation by starting with $f$ and correcting again and again.
For example, $\varphi(x) = 1 + \int_0^x \varphi(s) \, ds$ is of the second kind.
Starting with $1$ and correcting gives $1$, $1 + x$, $1 + x + x^2/2$, and so on, which builds up $e^x$.