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

CHAPTER 3. APPROXIMATE SOLUTIONS 57
For the first iteration,
At the second iteration,
After the third iteration,
f(0, y(0)) = (0, −2) (3.123)
y 1 = y 0 + hf(0, y(0)) (3.124)
= (1, 0) T + π 4 (0, −2)T ≈ (1, −1.5708) T (3.125)
t 1 = t 0 + h = π/4 (3.126)
f(t 1 , y 1 ) ≈ f(π/4, (1, −1.5708) T ) (3.127)
≈ (−1.5708, −2) T (3.128)
y 2 = y 1 + hf(t 1 , y 1 ) (3.129)
≈ (1, −1.5708) T + π (−1.5708, −2)
4
(3.130)
≈ (−.2337, −3.1416) (3.131)
t 2 = t 1 + π/4 = π/2 (3.132)
f(t 2 , y 2 ) ≈ f(π/2, (−.2337, −3.1416) T ) (3.133)
After the fourth and final iteration,
≈ (−3.1416, .4674) T (3.134)
y 3 = y 2 + hf(t 2 , y 2 ) (3.135)
≈ (−.2337, −3.1416) T + π 4 (−3.1416, .4674)T (3.136)
≈ (−2.7011, −2.7745) T (3.137)
t 3 = t 2 + π/4 = 3π/4 (3.138)
f(t 3 , y 3 ) ≈ f(3π/4, (−2.7011, −2.7745 T ) (3.139)
≈ (−2.7745, 5.4022) T (3.140)
y 4 = y 3 + hf(t 3 , y 3 ) (3.141)
≈ (−2.7011, −2.7745) T + π 4 (−2.7745, 5.4022)T (3.142)
≈ (−4.8802, 1.4684) T (3.143)
t 4 = t 3 + π/4 = π (3.144)
The Mathematica implementation discussed in section 1.8 does not even need to
be modified. For a scalar problem one would define a function f[t,y] that returns
a scalar value, and then call ForwardEuler[f, {t 0 , y 0 }, h, t max ]. The return
value is a list of the form
{{t 0 ,y 0 }, {t 1 , y 1 }, . . . }
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge