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

Chapter 4
Improving on Euler’s Method
4.1 The Test Equation and Problem Stability
Consider the initial value problem
y ′ = f(t, y), y(0) = y 0 (4.1)
What happens if we perturb the initial condition a small amount,
ŷ ′ = f(t, ŷ), ŷ(0) = ŷ 0 (4.2)
where ŷ 0 = y 0 + δ for some small number δ? How will the two solutions behave with
respect to one another?
Consider, for example, the following initial value problem, which we shall refer
to as the scalar test equation.
y ′ = λy, y(0) = y 0 (4.3)
We will find it instructive to test many of our methods with the scalar test problem
and will formulate much of our theory around this problem, mainly because it is
easy to solve and we can describe its behavior very easily.The solution of the scalar
problem is
y(t) = y 0 e λt (4.4)
The perturbed equation has solution
and so if λ ∈ R the two solutions will differ by
y(t) = ŷ 0 e λt (4.5)
ɛ(t) = |y(t) − ŷ(t)| = |y 0 − ŷ 0 |e λt (4.6)
Therefore
⎧
⎪⎨ 0, λ < 0
lim ɛ(t) = |y
t→∞ ⎪ 0 − ŷ 0 |, λ = 0
⎩
∞, λ > 0
69
(4.7)