#### CS3414 Afterclass Notes --- 11 June, 2002

Ordinary Differential Equations (parts of Chapters 8 and 12)
1. Boundary Value Problems (BVPs)

1. The (2nd-order) problem

x''(t) = f(t, x, x')
x(a) = alpha, x(b) = beta

2. 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.

3. 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)