Some problems in applied mathematics contain a small parameter eg.,e. Setting this parameter to 0 sometimes leads to an easier problem - called "unperturbed problem".
The solution of unperturbed problem(e=0) might be used as an approximation of the perturbed problem (e<>0). We then seek a series of corrections to the initial approximation and hope that these few additional terms provide a useful approximate solution to the perturbed problem.
For example consider the following algebraic equation:
F(x) : x^2 - 3.99*x + 3.02 =0
Using e=0.01, we obatain:
x^2 + (e-4)*x + (3+2*e) = 0
The unperturbed problem is:
x^2 - 4*x + 3 = 0
We know that unperturbed problem has solutions x1=3 and x2=1
To solve perturbed problem we seek asymptotic expansion of the solutions and finally obain the following expasions around the root:
x1(e) = 3-(5/2)*e+(25/4)*e*e
x2(e) = 1+(3/2)*e+(15/8)*e*e
Now to get solution of the original problem, we substitute e=0.01:
x1(0.01)=2.9748125
x2(0.01)=....
