The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro
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66 CHAPTER 3. APPROXIMATE SOLUTIONS<br />
3.8 Dependence Upon a Parameter<br />
<strong>The</strong>orem 3.11. (Dependence on Initial Conditions) Let f(t, y) ∈ C(R) where<br />
R = [t 0 − a, t 0 + a] × [ỹ 0 − b, ỹ 0 + b], (3.203)<br />
then there exists a unique solution of the IVP with perturbed IC,<br />
y ′ = f(t, y) (3.204)<br />
y(t 0 ) = y 0 (3.205)<br />
in the region<br />
|t − t 0 | ≤ h ′ (3.206)<br />
|y 0 − ỹ 0 | ≤ b/2 (3.207)<br />
|y − y 0 | ≤ Mh ′ (3.208)<br />
where<br />
( )<br />
h ′ b<br />
= min a,<br />
2M<br />
(3.209)<br />
|f(t, y)| ≤ M in R (3.210)<br />
Figure 3.4: <strong>The</strong> initial condition can move along the vertical axis and a solution<br />
exists throughout the inner box.<br />
(t 0 + a, ỹ 0 + b)<br />
R<br />
✈<br />
(t 0 , y 0 )<br />
✈<br />
(t 0 , ỹ 0 )<br />
(t 0 + h ′ ,<br />
ỹ 0 + b 2 )<br />
(t 0 + h ′ ,<br />
ỹ 0 − b 2 )<br />
(t 0 + a, ỹ 0 − b)<br />
Math 582B, Spring 2007<br />
California State University Northridge<br />
c○2007, B.E.<strong>Shapiro</strong><br />
Last revised: May 23, 2007