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

Chapter 8
Boundary Value Problems
8.1 Shooting
In this section the method of shooting will be illustrated with an example. Suppose
we are given the boundary value problem
It is easily verified that an exact solution is
y ′′ + 1 t y′ (t) − 1 t 2 y = 0 (8.1)
y(1) = y(2) = 1 (8.2)
y(t) = 2 + t2
3t
(8.3)
Our goal is to find this solution numerically. We will do this by converting the the
problem into an initial value problem. Let u = y, v = y ′ . Then
u ′ = v, u(1) = 1 (8.4)
v ′ = 1 t 2 u − 1 v, v(1) = c (8.5)
t
where c is the unknown initial condition for v(1) = y ′ (1). We can estimate a first
guess at c using 8.7 and the given boundary data with a forward approximation to
the derivative:
c (0) = v(1) (0) = u ′ (1) ≈
u(2) − u(1)
2 − 1
= 0 (8.6)
The estimate is very rough and could be quite far from the actual value. Next
we will use our Forward Euler method (we could replace this with any other IVP
method, but we will stick to Forward Euler because it is the simplest, and works
quite well for illustrating the problem) to solve the first IVP approximation to our
BVP:
u ′ = v, u(1) = 1 (8.7)
v ′ = 1 t 2 u − 1 t v, v(1)(0) = c (0) = 0 (8.8)
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