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

CHAPTER 8. BOUNDARY VALUE PROBLEMS 161
One way to improve shooting is called multiple shooting. In this technique
simple shooting is applied separately on each interval of the mesh. Let y n be the
exact solution of the initial value problem
y ′ n = f(t, y n ) (8.58)
y n (t n−1 ) = c n−1 (8.59)
on the interval [t n−1 , t n ]. Assuming that each of these exact solutions is known then
the exact solution of
y ′ = f(t, y ) (8.60)
g(y(a), y(b)) = 0 (8.61)
on [a, b] satisfies
⎧
y 1 (t, c 0 ) on [t 0 , t 1 ]
⎪⎨ y 2 (t, c 1 ) on [t 1 , t 2 ]
y(t) =
.
⎪⎩
y N (t, c N−1 ) on [t N−1 , t N ]
(8.62)
if the following patching conditions, which ensure continuity on the entire interval
[a, b], are met:
⎧
y 1 (t 1 ; c 0 ) = c 1
⎪⎨
y 2 (t 2 ; c 1 ) = c 2
.
(8.63)
y N−1 (t N−1 ; c N−2 ) = c N−1
⎪⎩
g(c 0 , y N (b, c N−1 )) = 0
The patching conditions give a vector c of Nm coefficients, where
c = ( c T 0 c T 1 · · · c T T
N−1)
(8.64)
which satisfy the condition 8.63. As before, we will write the condition 8.63 as
H(c) = 0, (8.65)
which is a nonlinear equation that needs to be solved for c.
For the linear BVP,
y ′ = A(t)y + q(t) (8.66)
B a y(a) + B b y(b) = d (8.67)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge