---
tag: TH4NGC4016
kind: concept
name: Integro-partial differential equation
msc: [45K05, 35R09]
specializes: [YRKW2KX2CW]
sources:
  - url: https://en.wikipedia.org/wiki/Integro-differential_equation
---

## Summary

An integro-partial differential equation is an equation for an unknown function of several variables.
It involves both partial derivatives of the function and integrals of it.
It combines a [[partial differential equation|GC5BDZ6QBC]] with integral terms, which typically describe nonlocal interactions in space or memory in time.
Main examples are the [[Boltzmann equation|MH6CSWD7F0]] of gas dynamics, the [[radiative transfer equation|EX9YYRMAJA]], [[population balance equations|6Q7H2FQ12Q]] and nonlocal diffusion equations.
Their analysis mixes PDE methods with the theory of integral operators.

## Explanations

### L1 human

In the heat equation, heat at a point flows only to nearby points, which is a statement about derivatives.
In a nonlocal model, heat can jump from a point to any other point, with a rate that depends on the distance.
Then the change in time is given by an integral over all other points, and the equation contains both a time derivative and a space integral.
For example, $\partial_t u(x, t) = \int_{\R} J(x - y) \big(u(y, t) - u(x, t)\big) \, dy$ is such a nonlocal diffusion equation.