Suppose there is an equation system

M(u) = v

which is not solvable, but there is a related system

L(u) = v

that is solvable; say

u = T(v)

Typically this means that L(u)=v is a linear system; i.e., L(u₁+u₂)=L(u₁+L(u₂). This means that also T(v₁+v₂)=T(v₁)+T(v₂).

The scheme of perturbation analysis is to express the unsolvable system as

M(u) = L(u) + (M(u)−L(u)) = v

and work with the deviation function N(u)= (L(u)−M(u)). The above equation is then

L(u) = v + N(u)

Then a scale parameter ε is introduced so the equation under analysis is

L(u) = v + εN(u)

A solution of the form

u = u₀ + εu₁ + ε²u₂ + …

is then sought.

If ε=0 then L(u)=v and hence u₀=T(v).

In general

L(u₀ + εu₁ + ε²u₂ + …) = v + εN(u₀ + εu₁ + ε²u₂ + …)

Because of the linearity of L( ) the LHS of the above is

L(u₀) + εL(u₁) + ε²L(u₂) + …

If N(u) is analytic in u then

N(u₀ + εu₁ + ε²u₂ + … ) = N(u₀) + εN₁(u₀, u₁) + ε²N₂(u₀, u₁, u₂ ) + …

If the LHS and RHS of the previous equation are equated the coefficients of the correspondin powers of ε must be equal. This means

L(u₀) = v
L(u₁) = N(u₀)
L(u₂) = N₁(u₀, u₀)
…

The solution is then recursive; i.e.,

u₀ = T(v)
u₁ = T(N(u₀)) = T(N(T(v))
u₂ = T(N₁(u₀, u₁)) = T(N₁(T(v), T(N(T(v)))
…

The solution to the original system is the general solution with ε set equal to 1.

Source:

Richard Bellman, Perturbation Techniques in Mathematics, Physics, and Engineering, Holt, Rhinehart and Winston, Inc., New York, 1964.