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

CHAPTER 3. APPROXIMATE SOLUTIONS 53
Integrate both sides from t 0 to t,
∫ t
d (
p(x)e
−Kx ) ∫ t
dx ≤ ɛ e −Kx dx
t 0
dx
t 0
(3.88)
p(t)e −Kt − p(t 0 )e −Kt 0
≤ − ɛ [
e −Kt − e −Kt ] 0
K
(3.89)
Solving for p(t) gives the fundamental inequality. The proof for p(t) < 0 is analogous.
Now suppose that p(t) = 0 identically for all t. Then the fundamental inequality is
equivalent to 0 < something positive and follows trivially.
3.4 Cauchy-Euler Existence Theory
In this section the quantities f, (t 0 , y 0 ), D, h are defined as before; R is the usual 2h
wide rectangle centered on (t 0 , y 0 ).
Theorem 3.5. Uniqueness. Suppose that f ∈ C(D), f ∈ L(y; K)(D), and let
(t 0 , y 0 ) ∈ D. Then there is at most one solution to 3.1.
Proof. Suppose that y(t) and ỹ(t) are exact solutions of 3.1. Then
From the fundamental inequality, equation (3.75),
y ′ (t) = f(t, y(t)), y(t 0 ) = y 0 (3.90)
ỹ ′ (t) = f(t, ỹ(t)), ỹ(t 0 ) = y 0 (3.91)
|y(t) − ỹ(t)| ≤ e K|t−t 0| |y(t 0 ) − ỹ(t 0 )| + ɛ 1 + ɛ 2
K
Hence the two functions are identical to one another for all t.
( )
e K|t−t0| − 1 = 0 (3.92)
Theorem 3.6. Fundamental Existence Theorem. Let f(t, y) ∈ L(y, K)(D).
Then an exact solution of the initial value problem 3.1 exists on some interval |t −
t 0 | < h.
Proof. Let ɛ n → 0 monotonically. Then by theorem 3.3 a sequence of ɛ n -approximate
solutions y n (t) exist, such that ⇐=
∣
∣y ′ n(t) − f(t, y n (t)) ∣ ∣ ≤ ɛ n , |t − t 0 | ≤ h (3.93)
except possibly at the finite set of points t (n)
i
, i = 1, ..., m n
By the fundamental inequality, for any integers n and p,
|y n (t) − y n+p (t)| ≤ e K|t−t0| |y n (t 0 ) − y n+p (t 0 )| + ɛ n + ɛ
( )
n+p
e K|t−t0| − 1 (3.94)
K
≤ e K|t−t0| |y 0 − y 0 | + 2ɛ ( )
n
e Kh − 1 → 0 (3.95)
K
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge