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

48 CHAPTER 3. APPROXIMATE SOLUTIONS
3.2 Epsilon-Approximate Solutions
To prove that a Euler’s method solution can actually be constructed we need to
formalize the definition of an approximate solution a little. 1
Definition 3.1. Let f(t, y) ∈ C 0 on some domain D ⊂ R 2 that contains the rectangle
R = [t 0 − a, t 0 + a] × [y 0 − b, y 0 + b] (3.38)
Then y(t) is an ɛ-approximate solution of the initial value problem 3.1
all of the following conditions are true:
if
1. y(t) is admissable on D, i.e., f(t, y(t)) ∈ D for all t ∈ [t 0 − a, t 0 + a];
2. y(t) is continuous on [t 0 − a, t 0 + a];
3. y ′ (t) exists except at finitely many points t 1 , t 2 , ..., t n and is continuous on
t i < t < t i+1 for all i;
4. |y ′ (t) − f(t, y(t)| ≤ ɛ for all t ∈ [t 0 − a, t 0 + a] except possibly at t 1 , t 2 , ....
Theorem 3.3. Let (t 0 , y 0 ), R, D and f be defined as in definition (3.1); furthermore,
suppose that f is bounded on R, namely that there exists some M > 0 such
that
|f(t, y)| ≤ M (3.39)
for all (t, y) ∈ R; and define
h = min(a, b/M) (3.40)
Then it is possible to construct an ɛ-approximate solution to (3.1).
Proof. We will construct the solution going to the “right”, namely, for t > t 0 . The
proof for the left-side solution is analogous.
Define the rectangle S ⊂ R by
Since
S = [t 0 − h, t 0 + h] × [y 0 − Mh, y 0 + Mh] (3.41)
h = min(a, b/M) ≤ a (3.42)
Mh = min(Ma, b) ≤ b (3.43)
then S ⊂ R. Hence all of the properties of f that hold on R also hold on S.
=⇒
Let ɛ > 0 be given. Then since f(t, y) is continuous, it is uniformly continuous
on S, and therefore there exists some δ > 0 such that
Press.
1 This discussion parallels W. Hurewicz (1958) Lectures on Ordinary Differential Equations, MIT
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007