---
tag: ZDS77Q0F06
kind: concept
name: Integral equation of the first kind
msc: [45A05, 45B05]
specializes: [AYZCKG34QK]
sources:
  - url: https://en.wikipedia.org/wiki/Integral_equation
---

## Summary

An integral equation of the first kind is a [[linear integral equation|AYZCKG34QK]] in which the unknown function appears only under the integral: $\int K(x, s) \varphi(s) \, ds = f(x)$.
Solving it means inverting an integral operator.
Such equations are typically ill-posed: a solution may fail to exist, and small changes of $f$ can cause large changes of $\varphi$.
Inverse problems, such as recovering a signal from blurred data, lead to equations of the first kind.

## Explanations

### L1 human

One only sees a smoothed version $f$ of the unknown $\varphi$, and one wants $\varphi$ back.
Smoothing destroys fine detail, so going back amplifies noise.
For example, $\int_0^x \varphi(s) \, ds = f(x)$ asks for a function whose integral is $f$.
Its continuous solution is $\varphi = f'$, which exists only when $f$ is continuously differentiable with $f(0) = 0$.