---
tag: 5FQG18RD00
kind: definition
name: Convolution integral equation
msc: ["45E10", 44A35]
defines: EY75YD988Z
uses: [R3SG1SN7JQ, B5AB451TB7, 2Z29JN3VES, SASP14MA1Z]
sources:
  - url: https://en.wikipedia.org/wiki/Convolution
---

## Definition

Let $k \in L^1(\R)$, let $f \colon \R \to \C$ be a given function, and let $\lambda \in \C$.
A *convolution integral equation of the second kind* on $\R$ is the equation
$$\varphi(x) - \lambda \int_{-\infty}^{\infty} k(x - t) \, \varphi(t) \, dt = f(x), \quad x \in \R,$$
for an unknown function $\varphi$.
A *convolution integral equation of the first kind* is the equation
$$\int_{-\infty}^{\infty} k(x - t) \, \varphi(t) \, dt = f(x), \quad x \in \R.$$
In both cases the kernel $(x, t) \mapsto k(x - t)$ is a [[difference kernel|B5AB451TB7]], and the integral is the [[convolution|2Z29JN3VES]] $(k * \varphi)(x)$.
The same names are used on $\R^n$, and on an interval such as $[0, \infty)$ when the integral runs only over that interval.

## Remarks

Example: take $k(s) = e^{-\abs{s}}$, $\lambda = \frac{1}{4}$ and $f(x) = \frac{1}{2} e^{-\abs{x}}$.
The Fourier transform of $e^{-a\abs{s}}$ is $\frac{2a}{a^2 + \xi^2}$ for $a > 0$, so the transformed equation reads $\hat\varphi(\xi) \left(1 - \frac{1}{2(1 + \xi^2)}\right) = \frac{1}{1 + \xi^2}$.
Hence $\hat\varphi(\xi) = \frac{2}{1 + 2\xi^2} = \frac{1}{1/2 + \xi^2}$, and $\varphi(x) = \frac{1}{\sqrt{2}} e^{-\abs{x}/\sqrt{2}}$.

Non-example: the equation $\varphi(x) - \int_{-\infty}^{\infty} e^{-x^2 - t^2} \varphi(t) \, dt = f(x)$ is a linear integral equation, but its kernel $e^{-x^2 - t^2}$ is not a function of $x - t$ alone, so it is not of convolution type.

The equation on the half-line $\varphi(x) - \int_0^\infty k(x - t) \varphi(t) \, dt = f(x)$, $x > 0$, is the [[Wiener–Hopf equation|QYY4NMDY5H]].
The Volterra form $\varphi(x) - \int_0^x k(x - t) \varphi(t) \, dt = f(x)$ is the [[Volterra equation of convolution type|SNFYTC1TTA]].