- 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.

- 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.
- 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. - Famous examples of finite difference formulas.
x(t+h) - x(t) x'(t) = ------------- + O(h) h

x(t+h) - x(t-h) x'(t) = --------------- + O(h^2) 2h

x(t-h) - 2x(t) + x(t+h) x''(t) = ----------------------- + O(h^2) h^2

- The error terms above (called `discretization' error, or
sometimes `truncation' error) are derived using Taylor series.
- The rest of the finite difference story: formulas can be derived
that ...
- achieve higher accuracy (requires more points)
- use unequally spaced points
- use points on only one side of t (`one-sided' formulas)

- A

- The (2nd-order) problem