# Euler's Method - Global Error

• Nov 24th 2011, 07:30 AM
MathSucker
Euler's Method - Global Error
Use Euler's method with $h = 1/2$ to estimate $y(1)$ for the IVP:

$y(0)=1$

$y'(t)=t^2-y(t)$

Assuming that $|y(t)| \le 1$ for $0 \le t \le 1$ determine the value of n needed to ensure that $|E_n| \le 10^{-2}$

The first part is easy enough:

$y_1=y_0+f(t_0,y_0)h=1+f(0,1)(1/2)=1/2$

$y_2=y_1+f(x_1,y_1)h=1/2-1/8=3/8$

$\Rightarrow y(1)=3/8$

I'm having trouble with the second part. I know I need to use:

$|E_n|\le \frac{T}{L}\left(e^{L(t_n-t_0)}-1\right)$

Could somebody explain to me how I find $L$ and $T$?
• Nov 24th 2011, 08:11 AM
MathSucker
Re: Euler's Method - Global Error
Is this correct for $L$:

$|f(t,u)-f(t,v)|=|t^2-u-t^2+v|=|u-v|$

Lipschitz with $L=1$
• Nov 25th 2011, 03:04 AM
MathSucker
Re: Euler's Method - Global Error
It's $T$ that I'm having trouble with... how would I find the upper bound for $|y''(t)|$?