## anonymous 5 years ago how does eulers method of approximation work?

1. nowhereman

There are two kinds of Euler approximations called explicit and implicit. The explicit is a bit faster but the implicit one is much more stable. What you do is approximating the solution to an initial-value problem by piece-wise affine functions. So explicit Euler is the easiest approximation method you can think of. Just calculate the elevation at the starting point, go a bit in that direction and take the resulting end point as the new starting point. Then repeat that step.

2. anonymous

and implicit?

3. amistre64

so instead of going to x=0 you just step down a bit?

4. nowhereman

Don't what you mean amistre... For implicit in each step you set up a system of linear equations which contains the elevation at the starting and at the end point.

5. anonymous

oh ok, thanks man

6. amistre64

elevation at the starting point..whats that mean? go abit in that direction and take the end point..... whats that mean?

7. amistre64

8. nowhereman

You have given a initial value problem $f'(t) = F(t, f(t)),\; f(t_0) = y_0$ So the starting point is (t_0, y_0). Also you must choose a step size h, which then gives you the precision. The elevation at the starting point is $f'(t_0) = F(t_0, y_0)$ So going in that direction you get the approximation $f(t_0 + h) \approx y_0 + h\cdot F(t_0, y_0)$

9. amistre64

ahhh... that sounds more like integration methods.... unless im mistaken :)

10. nowhereman

Well yes, solving initial-value problems / differential equations is a generalization of integration.

11. amistre64

i was thinking more along the lines of the Newton stuff in the books finding roots and what nots. ok

12. nowhereman

Hehe, that explains the confusion ;-)