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

CHAPTER 2. SUCCESSIVE APPROXIMATIONS 29
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To print the first 5 iterations of y ′ = y cos t, y(0) = 1 using this function, one enters
g[tvariable , yvariable ]:= yvariable*Cos[tvariable];
Picard[g, t, 0, 1, 5];
which prints
φ 0 =1
φ 1 =1+Sin[t]
φ 2 =1+Sin[t]+ 1 2 Sin[t]2
φ 3 =1+Sin[t]+ 1 2 Sin[t]2
+ 1 6 Sin[t]3
φ 4 =1+Sin[t]+ 1 2 Sin[t]2 + 1 6 Sin[t]3 + 1
24 Sin[t]4
φ 5 =1+Sin[t]+ 1 2 Sin[t]2 + 1 6 Sin[t]3 + 1
24 Sin[t]4 + 1
120 Sin[t]5
and returns the value
1+Sin[t]+ 1 2 Sin[t]2 + 1 6 Sin[t]3 + 1
24 Sin[t]4 + 1
120 Sin[t]5
It appears that the sequence is converging to the series
φ(t) =
∞∑
k=0
sin k (t)
k!
= e sin t
It is easily verified by separation of variables or direct substitution that this is, in
fact, the correct solution.
2.2 Fixed Point Theory
The gist of the proof of Algorithm 2.1 is that it is a form of fixed point iteration.
Anyone who has ever played with a pocket calculator by pressing the same button
over and over again has learnt something about fixed point iteration. For example,
suppose you have set your calculator to show your answers to 3 digits of accuracy
to the right of the decimal point. Then type the number 16 and repeatedly press
the √ . you will generate the following sequence (try it!):
16, 4, 2, 1.414, 1.189, 1.090, 1.044, 1.021, 1.010, 1.005, 1.002, 1.001, 1.000, 1.000, 1.000, ...
A second example starts with the number 1 and uses the cos key. This time
the sequence of numbersis:
1., 0.540, 0.857, 0.654, 0.793, 0.701, 0.763, 0.722, 0.750, 0.731, 0.744, ...
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge