Page 164 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

154 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
Equation (10.2) can be expanded for small values of ε in the rapidly convergent series

ε ε 2 ε 4
x = 1 − + − + ··· (10.3)
2 8 128
or
ε ε 2 ε 4
x =−1 − − + − ··· (10.4)
2 8 128
To apply perturbation theory we first note that if ε = 0 the two roots of the equation, which
we will call the zeroth-order solutions, are x 0 =±1. We assume a solution of the form

2 3 4
x = x 0 + a 1 ε + a 2 ε + a 3 ε + a 4 ε + ··· (10.5)
Substituting (10.5) into (10.1)

2 2 3
1 + (2a 1 + 1)ε + a + 2a 2 + a 1 ε + (2a 1 a 2 + 2a 3 + a 2 )ε + ··· − 1 = 0 (10.6)
1
n
where we have substituted x 0 = 1. Each of the coefficients of ε must be zero. Solving for a n
we find
1
a 1 =−
2
1
a 2 = (10.7)
8
a 3 = 0

so that the approximate solution for the root near x = 1is

ε ε 2
4
x = 1 − + + O(ε ) (10.8)
2 8
4
The symbol O(ε ) means that the next term in the series is of order ε 4
Performing the same operation with x 0 =−1

2
2
3

1 − (1 + 2a 1 )ε + a − 2a 2 + a 1 ε + (2a 1 a 2 − 2a 3 + a 2 )ε + ··· − 1 = 0 (10.9)

1
n
Again setting the coefficients of ε equal to zero
1
a 1 =−
2
1 (10.10)
a 2 =−
8
a 3 = 0