# Convergence of Eulers method.. Differential equations

• Mar 8th 2008, 11:29 AM
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, 11:12 AM
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 \$\displaystyle kh^2\$, for some \$\displaystyle k\$ (depending on the function and the interval over which we are integrating) and the number of steps will be \$\displaystyle a/h\$ where \$\displaystyle a\$ is the length of the interval we are integrating over. So the total error at a fixed point will be bounded by \$\displaystyle kah\$, and so will go to zero as \$\displaystyle h\$ goes to zero.

RonL