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

CHAPTER 4. IMPROVING ON EULER’S METHOD 71
At t = t 0 + 2h the solution jumps again by δ, to
y(t 0 + 2h) = y 0 e 2λh + δe λ(t 0+2h−(t 0 +h))
(4.18)
= y 0 e 2λh + δe λh) (4.19)
hence for t 0 + h ≤ t < t 0 + 2h, we have
y(t 0 + 2h) = (y 0 e 2λh + δe λh + δ)e λ(t−(t 0+2h))
= y 0 e λ(t−t 0) + δe λ(t−(t 0+h)) + δe λ(t−(t 0+2h))
(4.20)
(4.21)
In general, the solution in the n th interval will be
y = y 0 e λ(t−t0) + δe λ(t−(t0+h)) + δe λ(t−(t0+2h)) + · · · + δe λ(t−(t 0+n))
(4.22)
n∑
= y 0 e λ(t−t0) + δe λ(t−t 0)
e −kλh (4.23)
Summing the geometric series,
k=1
y = y 0 e λ(t−t 0) + δe λ(t−t 0) 1 − e−nλh
e λh − 1
(4.24)
Figure 4.1: Illustration of the experimental process of simulating numerical error
due to numerical integration by a stepwise approach. The numerical solution is
illustrated by the black dots. Exact solutions to the IVP pass through those dots.
Starting at (t 0 , y 0 ), the exact solution is followed through (t 0 + h, y(t 0 + h)). Then a
new initial value problem is solved, starting at t 0 +h, with initial value y(t 0 +h)+δ.
The process is then repeated. The solution that emulates the numerical solution is
shaded.
.
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge