Ordinary Differential Equations (parts of Chapters 8 and 12)
Boundary Value Problems (BVPs)
The (2nd-order) problem
x''(t) = f(t, x, x')
x(a) = alpha, x(b) = beta
Shooting method
Idea: replace BVP by a series of 2nd-order IVPs, with a guess
for x'(a); iterate until the computed x(b) is (close enough) to
the prescribed value beta.
Important special case: if f(t,x,x') is linear in x and x'
then only two shots are necessary.
More formally, the idea is to find a root of the equation
phi(z) - beta = 0, where z is a guess for x'(a), and phi(z) is
the computed value of x(b) that results from solving the
corresponding IVP.
A method based on finite differences
A finite difference formula gives an approximation to
a derivative of a function x(t) using only a few evaluations of
that function at points near t.