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

CHAPTER 3. APPROXIMATE SOLUTIONS 55
where f(t, y) ∈ L(y; K)(D). Let ỹ(t) be an ɛ-approximate solution to y(t) satisfying
ỹ ′ (t 0 ) = y 0 for |t − t 0 | ≤ h. Then there exists a constant N > 0, independent of ɛ,
such that
|ỹ(t) − y(t)| ≤ ɛN (3.105)
for |t − t 0 | ≤ h.
Proof. From the fundamental inequality,
|ỹ(t) − y(t)| ≤ e K|t−t0| |ỹ(t 0 ) − y(t 0 )| + ɛ + 0
K
≤ e Kh |y 0 − y 0 | + ɛ ( )
e Kh − 1
K
≤ ɛ ( )
e Kh − 1
K
( )
e K|t−t0| − 1
(3.106)
(3.107)
(3.108)
So if we let
N = 1 K
( )
e Kh − 1
(3.109)
the conclusion to the theorem follows immediately.
Theorem 3.8. Let D be an arbitrary bounded domain, f ∈ L(y; K)(D). Let y(t)
be an exact solution of the IVP
y ′ = f(t, y), y(t 0 ) = y 0
that is also defined for t = t 1 . The the value of y(t 1 ) may be determined to any
desired degree of accuracy, using Euler’s method, but using a sufficiently small step
size.
Proof. Let d be the minimum distance between the curve y(t) and the boundary of
D. If η is any number 0 < η < d then the strip
S = {(t, y)|t 0 ≤ t ≤ t 1 , |y − y(t)| ≤ η} (3.110)
lies entirely in D.Let E be the desired error. Choose ɛ so that
(
)
Kη
ɛ = min E,
e K|t 1−t 0 | − 1
By equation 3.108
|ỹ(˜t) − y(˜t)| ≤ ɛ K
( )
e K|˜t−t 0 | − 1
(3.111)
(3.112)
≤ η eK|t 1−t 0 | − 1
e K|˜t−t 0 | − 1
(3.113)
< η (3.114)
Thus ỹ(t) is within the strip S. Within S, the conditions of the Fundamental
Inequality are met, so we can continue to approximate y by ỹ.Hence ỹ(t) is defined ⇐=
up to at least t = t 1 with the desired error.
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge