Page 166 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

156 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
and the equation is said to be singular as ε → 0. If we set xε = u we find an equation for u as

2
u + u − ε = 0 (10.16)
With ε identically zero, u = 0or −1. Assuming that u may be approximated by a series like
(10.5) we find that
2 2 3
(−a 1 − 1)ε + a − a 2 ε + (2a 1 a 2 − a 3 )ε + ··· = 0 (10.17)
1
a 1 =−1

a 2 = 1 (10.18)
a 3 =−2

so that
1
2
x =− − 1 + ε − 2ε + ··· (10.19)
ε
The three-term approximation of the negative root is therefore x =−10.92, within 0.03% of
the exact solution.
As a third algebraic example consider
2
x − 2εx − ε = 0 (10.20)
This at first seems like a harmless problem that appears at first glance to be amenable to a regular
2
perturbation expansion since the x term is not lost when ε → 0. We proceed optimistically
by taking

2 3
x = x 0 + a 1 ε + a 2 ε + a 3 ε + ··· (10.21)
Substituting into (10.20) we find

2
2
2

x + (2x 0 a 1 − 2x 0 − 1)ε + a + 2x 0 a 2 − 2a 1 ε + ··· = 0 (10.22)
0
1
from which we find
x 0 = 0
2x 0 a 1 − 2x 0 − 1 = 0 (10.23)
2
a + 2x 0 a 2 − 2a 1 = 0
1
From the second of these we conclude that either 0 =−1 or that there is something wrong.
That is, (10.21) is not an appropriate expansion in this case.
Note that (10.20) tells us that as ε → 0, x → 0. Moreover, in writing (10.21) we have
essentially assumed that ε → 0 in such a manner that x → constant. Let us suppose instead
ε