Lecture Notes in Differential Equations - Bruce E. Shapiro

94 LESSON 11. PICARD ITERATION
in which the Method of Successive Approximations converges to the unique
solution of the initial value problem
⎫
dy
= f(t, y) ⎬
dt
⎭
y(t 0 ) = y 0
(11.20)
The proof of this theorem is quite involved and will be discussed in the
sections 12 and 13.
The procedure for using the Method of Successive Approximations is summarized
in the following box. 1
Procedure for Picard Iteration
To solve y ′ = f(t, y) with initial condition y(t 0 ) = y 0 :
1. Construct the first 3 iterations φ 0 , φ 1 , φ 2 , φ 3 .
2. Attempt to identify a pattern; if one is not obvious you
may need to calculate more φ n .
3. Write a formula for the general φ n (t) from the pattern.
4. Prove that when you plug φ n (t) into the right hand side of
equation (11.15) you get the same formula for φ n+1 with
n replaced by n + 1.
5. Prove that φ(t) = lim n→∞ φ n converges.
6. Verify that φ(t) solve the original differential equation and
initial condition.
1 The Method of Successive Approximations is usually referred to as Picard iteration
for Charles Emile Picard (1856-1941) who popularized it in a series of textbooks on
differential equations and mathematical analysis during the 1890’s. These books became
standard references for a generation of mathematicians. Picard attributed the method
to Hermann Schwartz, who included it in a Festschrift honoring Karl Weierstrass’ 70’th
birthday in 1885. Guisseppe Peano (1887) and Ernst Leonard Lindeloff (1890) also
published versions of the method. Since Picard was a faculty member at the Sorbonne
when Lindeloff, also at the Sorbonne, published his results, Picard was certainly aware
of Lindeloff’s work. A few authors, including Boyce and DiPrima, mention a special
case published by Joseph Liouville in 1838 but I haven’t been able to track down the
source, and since I can’t read French, I probably won’t be able to answer the question
of whether this should be called Liouville iteration anytime soon.