The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 2. SUCCESSIVE APPROXIMATIONS 39
Where K is any number larger than sup (a,b) f y . If we choose the endpoints a and
b such that |b − a| < 1/K we have K|b − a| < 1. Thus T is a contraction. By the
contraction mapping theorem it has a fixed point; call this point φ. Equation 2.73
follows immediately.
Theorem 2.12 (Error Bounds on Picard Iterations). Under the same conditions
as before, let φ n be the n th Picard iterate, and let φ be the solution of the IVP. Then
|φ(t) − φ n (t)| ≤ M |K(t − t 0)| n+1
e K|t−t 0|
K(n + 1)!
(2.80)
where M = sup D |f(t, y)| and K is a Lipshitz constant. Furthermore, if L = |b − a|
then
‖φ(t) − φ n (t)‖ ≤ M [KL]n+1 e KL
(2.81)
K(n + 1)!
where ‖ · ‖ denotes the sup-norm.
Proof. We begin by proving the conjecture
For n = 1, equation 2.82 says that
|φ n − φ n−1 | ≤ Kn−1 M
|t − t 0 | n (2.82)
n!
|φ 1 − y 0 | ≤ M|t − t 0 | (2.83)
which follows immediately from equation 2.73. Next, make the inductive hypothesis
2.82 and calculate
∫ t
|φ n+1 − φn| =
∣ [f(s, φ n (s)) − f(s, φ n−1 (s))] ds
∣ (2.84)
t 0
≤ K
∫ t
t 0
|φ n (s) − φ n−1 (s)| ds (2.85)
by the definition of φ n and the Lipshitz condition. Applying the inductive hypothesis
and then integrating,
|φ n+1 − φn| ≤ Kn M
n!
which proves conjecture 2.82. Now let
≤
φ n (t) = φ 0 (t) +
∫ t
t 0
|s − t 0 | n ds (2.86)
Kn M
(n + 1)! |t − t 0| n+1 (2.87)
n∑
[φ i (t) − φ i−1 (t)] (2.88)
Then since the sequence of Picard iterates converges to the solution,
i=1
φ(t) = lim
n→∞ φ n(t) = φ 0 (t) +
∞∑
[φ i (t) − φ i−1 (t)] (2.89)
i=1
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge