Page 165 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

INTRODUCTION TO PERTURBATION METHODS 155
so that the root near x 0 =−1is

ε ε 2
4
x =−1 − − + O(ε ) (10.11)
2 8
The first three terms in (10.8) give x = 0.951249219, accurate to within 1.16% of the exact
value while (10.11) gives the second root as x =−1.051249219, which is accurate to within
1.05%.
Next suppose the small parameter occurs multiplied by the squared term,

2
εx + x − 1 = 0 (10.12)
Using the quadratic formula gives the exact solution.

1 1 1
x =− ± + (10.13)
2ε 4ε 2 ε
If ε = 0.1 (10.13) gives two solutions:

x = 0.916079783

and

x =−10.91607983

We attempt to follow the same procedure to obtain an approximate solution. If ε = 0 identically,
x 0 = 1. Using (10.5) with x 0 = 1 and substituting into (10.12) we find

2
2
3

(1 + a 1 )ε + (2a 1 + a 2 )ε + 2a 2 + a + a 3 ε + ··· = 0 (10.14)
1
n
Setting the coefficients of ε = 0,solvingfor a n , and substituting into (10.5)
2 3
x = 1 − ε + 2ε − 5ε + ··· (10.15)
gives x = 0.915, close to the exact value. However Eq. (10.12) clearly has two roots, and the
method cannot give an approximation for the second root.
The essential problem is that the second root is not small. In fact (10.13) shows that as
1 2
ε → 0, |x| → so that the term εx is never negligible.
2ε

10.1.2 Singular Perturbation
Arranging (10.12) in a normal form
x − 1
2
x + = 0 (10.12a)
ε