Solutions to Exercises
69
Solutions to Exercises 1
Exercise I. A. The Picard method is an iterative process determining the solution
of Eq. (2.2) from the repeated action of Eq. (2.9):
u
(n+1) (k, r) = C 0 F 0 ,η 0 (k 0 r 0 ) + C 0 k 0 F
0 ,η 0
(k 0 r 0 ) (r − r 0 )
+
r
r 0
(r − r
)L (k, r
)u
(n) (k, r
) dr
,
(2.202)
where n ≥ 0 and u (0) (k, r) = 0. As one imposes C 0 = k
− 0 −1
0
C 0 (η 0 ) −1 ,
Eq. (2.37) implies that C 0 F 0 ,η 0 (k 0 r) is an entire power series in k 2 . Therefore,
u (n) (k, r) is well defined in the whole complex k-plane.
One will show that u (n) (k, r) → u(k, r) for n → +∞. An immediate
recurrence in Eq. (2.202) implies that ∀n ≥ 1:
|u
(n+1) (k, r) − u
(n) (k, r)| =
r
r 0
(r − r
(1) )L (k, r
(1) ) . . .
×
r (n−1)
r 0
(r − r
(n) )L (k, r
(n) )[u
(1) (k, r
(n) ) − u
(0) (k, r
(n) )] dr
(n) . . . dr
(1)
≤ r
n
||L ||
n
||u
(1)
− u
(0)
|| ∞
r
r 0
. . .
r (n−1)
r 0
dr
(n) . . . dr
(1)
≤
r 2n ||L || n
n!
||u
(1)
− u
(0)
|| ∞ ,
(2.203)
where the norm ||L || is defined as:
||L || =
+ 1)
r 2
0
+ ||v l || ∞ + |k|
2 ,
(2.204)
and all supremum norms are defined for r 0 ≤ r ≤ R (same for r’). Equation
(2.203) implies that:
n
|u
(n+1) (k, r) − u
(n) (k, r)| ≤ exp(r
2
||L ||) ||u
(1)
− u
(0)
|| ∞ .
(2.205)
1 The input files, codes and code user manual associated to computer-based exercises can be found
at https://github.com/GSMUTNSR.