The method of multiple scales is a perturbation theory technique in which a solution is allowed to depend on several independent
time or space variables that describe different scales. For example, with a small
parameter 
, one may introduce
 |  |  |
(1)
|
 |  |
(2)
| |
 |  |
(3)
|
One continues similarly at higher orders and seeks a solution 
. The chain rule
then replaces 
by 
.
Choosing the dependence on the slow variables to remove secular terms gives approximations
that remain valid on longer intervals than an ordinary regular perturbation asymptotic
expansion.
