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Suppose there is an equation system
which is not solvable, but there is a related system
that is solvable; say
Typically this means that L(u)=v is a linear system; i.e., L(u_{1}+u_{2})=L(u_{1}+L(u_{2}). This means that also T(v_{1}+v_{2})=T(v_{1})+T(v_{2}).
The scheme of perturbation analysis is to express the unsolvable system as
and work with the deviation function N(u)= (L(u)−M(u)). The above equation is then
Then a scale parameter ε is introduced so the equation under analysis is
A solution of the form
is then sought.
If ε=0 then L(u)=v and hence u_{0}=T(v).
In general
Because of the linearity of L( ) the LHS of the above is
If N(u) is analytic in u then
If the LHS and RHS of the previous equation are equated the coefficients of the correspondin powers of ε must be equal. This means
The solution is then recursive; i.e.,
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.
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