---
tag: 4B7M4A1MH0
kind: concept
name: Partial difference equation
msc: [39A14]
sources:
  - url: https://en.wikipedia.org/wiki/Recurrence_relation
---

## Summary

A partial difference equation is an equation for an unknown function of two or more integer variables.
It relates the values of the function at a point and at finitely many shifted points.
It is to an ordinary difference equation what a [[partial differential equation|GC5BDZ6QBC]] is to an ordinary differential equation.
Such equations arise from discretizing partial differential equations on a grid, from counting problems in combinatorics, and from lattice models in physics.

## Explanations

### L1 human

Fill the squares of an infinite grid with numbers by a local rule.
Each value is tied to the values at a few neighbouring squares.
Pascal's triangle is the classic case: each binomial coefficient is the sum of the two above it, $\binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}$.
A computer that simulates heat flow on a grid also solves a partial difference equation at every time step.