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

70 CHAPTER 4. IMPROVING ON EULER’S METHOD
In particular, the difference is bounded is λ ≤ 0, and we say that the differential
equation is stable. If λ < 0 and the difference approaches zero, we say that the
differential equation is asymptotically stable.
If λ ∈ C then the same argument holds for Re(λ), since
We can formally define stability as follows.
ɛ(t) = |y(t) − ŷ(t)| = |y 0 − ŷ 0 |e Re(λ)t (4.8)
Definition 4.1 (Stability). A solution y(t) is stable if, for any ɛ > 0 there is a
δ > 0 such that all other solutions ŷ(t) satisfying the same IVP with
|y 0 − ŷ 0 | ≤ δ (4.9)
also satisfies
for all values of t ≥ 0.
|y(t) − ŷ(t)| ≤ ɛ (4.10)
Definition 4.2 (Absolute Stability). A solution is absolutely stable if it is
stable and
lim |y(t) − ŷ(t)| = 0 (4.11)
t→∞
Definition 4.3 (Totally Stable). Let
y ′ = f(t, y) + δ(t), y(t 0 ) = y 0 + δ (4.12)
define a perturbation of the initial value problem 4.1. Let ŷ and y be the solutions
corresponding to two different perturbations {δ(t), δ} and {ˆδ(t), ˆδ}. Then the IVP
is said to be totally stable if there exists a positive constant S such that for all
t ∈ [a, b], whenever
|δ(t) − ˆδ(t)| ≤ ɛ and |δ − ˆδ| ≤ ɛ (4.13)
then
|y − ŷ| ≤ Sɛ (4.14)
Let us examine what this means for a numerical solution. Suppose we are solving
the test equation numerically in steps of size h; at the end of each step, an additional
error of δ accumulates. We can “simulate” this process by integrating the solution
exactly on each interval, and then perturbing the next initial condition by δ as
illustrated in figure 4.1. For the first interval, the calculated solution is
y = y 0 e λ(t−t 0) , t 0 ≤ t < t 0 + h (4.15)
At t = t 0 + h the solution by δ, hence for t 0 + h ≤ t < t 0 + 2h, we have
y = (y 0 e λ(t 0+h−t 0 ) + δ)e λ(t−(t 0+h))
= y 0 e λ(t−t 0) + δe λ(t−(t 0+h))
(4.16)
(4.17)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007