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

CHAPTER 8. BOUNDARY VALUE PROBLEMS 159
Theorem 8.1. If the matrix Q = B a Y (a) + B b Y (b) is invertible then the solution
to
y ′ = A(t)y + q(t) (8.42)
B a y(a) + B b y(b) = d (8.43)
exists and is given by
[
y(t) = Y (t) c +
∫ t
c = Q −1 [d − B b Y (b)
a
]
Y −1 (s)q(s)ds
∫ b
a
]
Y −1 (s)q(s)ds
(8.44)
(8.45)
If we define the matrix Φ(t) = Y (t)Q −1 then
y(t) = Φ(t) +
∫ b
a
G(t, s)(q)(s)ds (8.46)
where G(t, s) is the Green’s function of the differential equation
{
Φ(t)B a φ(a)Φ −1 (s), s ≤ t
G(t, s) =
−Φ(t)B b Φ(b)Φ −1 (s), s > t
(8.47)
8.3 Shooting Revisited
We return to solving the general boundary value problem
y ′ = y(t, y) (8.48)
g(y(a; c), y(b; c)) = 0 (8.49)
where we have used the notation y(t, c) to indicate the solution at t that satisfies
the initial condition y(a; c) = c. The we define the function
H(c) = g(c, y(b, c)) = 0 (8.50)
To solve this equation for c we can iterate using Newton’s method
c i+1 = c i −
( ) ∂H −1
H(c i ) (8.51)
∂c
Using the notation of 8.17, where g = g(u, v), by the chain rule
∂H
∂c = ∂H ∂u
∂u ∂c + ∂H ∂v
∂v ∂c
(8.52)
= B a Y (a) + B b Y (b) (8.53)
= Q (8.54)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge