# Thread: Euler's implicit formula for diff. eqs. - Convergence

1. ## Euler's implicit formula for diff. eqs. - Convergence

Hi good people, I'm doing some old exam questions and here's one I don't quite get.

We have the ordinary diff. eq,

$y'(t) = e^{y(t)}$

where $y(0)=1$ and $t\in [0,1]$.

If we want to solve this by using Euler's implicit formula, we need solve an equation by say fixed-point iteration in each step. How small must the step size $h$ be in order for the fixed-point iteration to converge on each step.

First of all, the problem does not say what kind of implicit formula I need to use. I get to the formula in the following manner,

$\int^{t_{i+1}}_{t_i} y'(t)dt = \int^{t_{i+1}}_{t_i}f(t,y(t))dt$,

and so,

$y(t_{i+1}) = y(t_i) + \int^{t_{i+1}}_{t_i}f(t,y(t))dt$.

To get an implicit formula from this I could simply approximate the integral by,

$f(t_{i+1},y(t_{i+1}))(t_{i+1}-t_i) = e^{y(t_{i+1})}h$,

or I could approximate it by the Trapezoidal rule.. Let's assume I do the former such that,

$y(t_{i+1}) = y(t_i) + e^{y(t_{i+1})}h.$

Now I need to use fixed-point iteration to solve this equation. If I let my first guess at $y(t_{i+1})$ be $y(t_i)$ which I believe is a reasonable guess, then I could write this thing in the familiar way;

$x_{n+1} = x_n + e^{x_n}h = g(x_n)$.

The sequence defined by the expression above converges if $0\leq|g'(x)|<1$ for all $x$ in the interval. But $|g'(x)|=|1+e^xh|<1$ and so $h<0$ but that doesn't make much sense does it? I don't think the step size must be zero Hope someone can help me out here.

Thanks.

2. This DE is solvable exactly...

$\displaystyle \frac{dy}{dt} = e^{y}$

$\displaystyle e^{-y}\,\frac{dy}{dt} = 1$

$\displaystyle \int{e^{-y}\,\frac{dy}{dt}\,dt} = \int{1\,dt}$

$\displaystyle \int{e^{-y}\,dy} = t + C_1$

$\displaystyle -e^{-y} + C_2 = t + C_1$

$\displaystyle -e^{-y} = t + C$ where $\displaystyle C = C_1 - C_2$

$\displaystyle e^{-y} = -t - C$

$\displaystyle -y = \ln{(-t - C)}$

$\displaystyle y = -\ln{(-t-C)}$

3. Ah, I did not notice that. Thanks.
But it should still be solvable by Euler's implicit formula and so my argument must have some kind of mistake I think..
Anyone see where I went wrong?