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

54 CHAPTER 3. APPROXIMATE SOLUTIONS
as n → ∞. Hence the sequence y n (t) converges uniformly to a continuous function
y(t).
Consequently, y n (t) → y(t) uniformly, and since f(t, y) is continuous on a rectangle
R, the sequence f(t, y n (t)) → f(t, y(t)) uniformly on R. Hence the sequence of
integrals
uniformly on R.
∫ t
t 0
f(x, y n (x))dx →
∫ t
t 0
f(x, y(x))dx (3.96)
We now integrate both sides of inequality 3.93 from t 0 to t.
∫ t
[ ]
∫ dyn (x)
t
∣[ ∣∣∣
∣
− f(x, y n (x)) dx
t 0
dx
∣ ≤ dyn (x)
∣∣∣
− f(x, y n (x))]∣
dx (3.97)
t 0
dx
∫ t
≤ ɛ n dx (3.98)
t 0
≤ ɛ n |t 0 − t| (3.99)
≤ ɛ n h (3.100)
By the fundamental theorem of calculus, since the y n are continuous, we can also
integrate the first term on the left side of the inequality,
∫ t
∣ y n(t) − y n (t 0 ) − f(x, y n (x))dx
∣ ≤ ɛ nh (3.101)
t 0
Taking the limit of both sides of the equation as n → ∞,
∫ t
y(t) = y(t 0 ) + f(x, y(x))dx (3.102)
t 0
We observe that this function satisfies the initial value problem. Hence a solution
exists.
When we defined the ɛ-approximate solution we only required that it satisfy
the initial value problem, namely, the differential equation and the initial condition.
We did not set any restrictions on whether or not the ”approximation” matched
the real solution in any way whatsoever. The following theorem tells us that we
can construct an ɛ-approximation that is arbitrarily close to the actual solution. In
other words, it is possible to use this technique to find a solution to the IVP that
matches the true solution with any arbitray accuracy.
Theorem 3.7. Cauchy-Euler Approximation Theorem. Suppose that y(t) is
an exact solution of the initial value problem
y ′ = f(t, y) (3.103)
y ′ (t 0 ) = y 0 (3.104)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007