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Thread: Euler's Method - Global Error

  1. #1
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    Euler's Method - Global Error

    Use Euler's method with $\displaystyle h = 1/2$ to estimate $\displaystyle y(1)$ for the IVP:

    $\displaystyle y(0)=1$

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

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

    The first part is easy enough:

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

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

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

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

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

    Could somebody explain to me how I find $\displaystyle L$ and $\displaystyle T$?
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  2. #2
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    Re: Euler's Method - Global Error

    Is this correct for $\displaystyle L$:

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

    Lipschitz with $\displaystyle L=1$
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  3. #3
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    Re: Euler's Method - Global Error

    It's $\displaystyle T$ that I'm having trouble with... how would I find the upper bound for $\displaystyle |y''(t)| $?
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