MAT360 Section Summary: 5.3

Higher-Order Taylor Methods

1. Summary

   Rather than stop at the first term in the Taylor expansion, as Euler did,

   

   we continue on and create an order Taylor method:

   

   You're perhaps wondering how we're going to compute the higher derivatives of y: well, recall the chain rule that we thought might come in handy sometimes for bounding the second derivatives in Euler's error calculations:

   

   or, more simply,

   

   We can continue this for higher derivatives, although the results quickly look rather nasty: e.g.

   

   It's actually a lot easier if you're working with a particular case and don't have to work in general. For example, if you were looking at Exercise 6b, p. 256,

   

   Then f(t,y)=t+y, and all higher partial derivatives of f disappear: so the general form is wasteful. We simply compute higher derivatives directly, as follows:

   

   

   So we've figured out quickly that all higher derivatives of y are equal. This is an interesting development: it means that y is a function of the form (which is its own derivative plus some ``transiant stuff'' that disappeared quickly from the higher derivatives (sounds like a polynomial to me...)). You can check that the general solution is

   

   where . For 6b, p. 256, , so y(t)=1+t is the unique solution.

2. Definitions
   • Local Truncation Error: The error made in approximating the solution of an IVP with a difference scheme of the form

     

     has local truncation error

     

     It's the error we'd make at using the particular scheme.

   For Euler's method, the local truncation error is

   

   for some .

   For the Taylor method of order 2,

   

   so the local truncation error is

   

   for some .

3. Theorems/Formulas

   Theorem 5.12: If Taylor's method of order n approximates the usual IVP

    

   and if , then the local truncation error is .