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The Scheme of Perturbation Analysis

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.


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

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