# Convergence of Eulers method.. Differential equations

• Mar 8th 2008, 12:29 PM
mathlete2
Convergence of Eulers method.. Differential equations
Prove that the approximation of Euler's method converges to the exact solution at any fixed point as h->0.
- Because h->0 we know that the number of intervals h approaches infinity. I just can't seem to come up with a way of proving this theorem using algebra.
• Mar 9th 2008, 12:12 PM
CaptainBlack
Quote:

Originally Posted by mathlete2
Prove that the approximation of Euler's method converges to the exact solution at any fixed point as h->0.
- Because h->0 we know that the number of intervals h approaches infinity. I just can't seem to come up with a way of proving this theorem using algebra.

Look at the error analysis, the error per step should be bounded above by $kh^2$, for some $k$ (depending on the function and the interval over which we are integrating) and the number of steps will be $a/h$ where $a$ is the length of the interval we are integrating over. So the total error at a fixed point will be bounded by $kah$, and so will go to zero as $h$ goes to zero.

RonL