Determine n for Ensuring Global Error is Less Than 10^-2 in Euler's Method

[spitz]

Homework Statement

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}

Homework Equations

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

The Attempt at a Solution

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. Could somebody help me out?

 

[spitz]

Is this correct for L:

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

Lipschitz with L=1

 

[spitz]

Please, Math Gods, I beg thee...

 

[HallsofIvy]

Is this correct for L:

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

Lipschitz with L=1

Yes, that's true.

 

[spitz]

Thanks, it's T that I'm having trouble with... how would I find the upper bound for |y''(t)| ?

 

[spitz]

Is this right?

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

So, T=(1/2)*(1/2)*y''(1)=(1/4)(2(1)-1^2-3/8)=0.15625

and

E_n\le|0.15625(e-1)|\Rightarrow E_n\le 0.26848

Then set E_n=0.01

\Rightarrow 0.01 = h(0.53696)
h=0.018623361
\frac{1-0}{n}=0.018623361
\Rightarrow n ≈ 54

Anyone? Anyone?