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