---
tag: 579J5K9K5Z
kind: concept
name: Similarity solution
msc: [35C06, 35B06, 76M55]
sources:
  - url: https://en.wikipedia.org/wiki/Self-similar_solution
same_as:
  wikidata: Q4056366
  wikipedia: Self-similar solution
---

## Summary

A similarity solution, or self-similar solution, of a partial differential equation is a solution that is unchanged by a scaling of its variables.
Its graph at one time is a rescaled copy of its graph at any other time.
Such a solution has the form $u(x, t) = t^{-\alpha} v(x / t^{\beta})$, so finding it reduces to an equation in fewer variables.
Many fundamental solutions, [[rarefaction waves|MFN4Q5SH8H]] and blow-up profiles are of this kind.
See the [[definition|3DJMFT0SQM]].

## Explanations

### L1 human

Drop ink into still water: the blot grows and fades, but its shape stays the same, only stretched.
A similarity solution is a solution with this property.
If one measures distance in units that grow like a power of time, the picture stops changing.
For the heat equation the blot has width proportional to $\sqrt{t}$ and height proportional to $t^{-1/2}$ in one dimension.
The variable $x / \sqrt{t}$ is then the only one that matters, and the PDE becomes an ordinary differential equation.