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\noindent{\large {\bf MATH 269A} {\bf HW \#3} (due on Friday, October 26)}

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\noindent{\bf [1]} Consider $\Phi=\Phi(x,y,h)$ associated with the 2nd 
order R-K methods: 
$$\Phi(x,y,h)=\frac{1}{2}\Big[f(x,y)+f(x+h,y+hf(x,y))\Big]$$
$$\Phi(x,y,h)=f(x+\frac{h}{2},y+\frac{h}{2}f(x,y)).$$
Verify that, under appropriate assumptions on $f$, we have: 
$$\frac{\partial \Phi}{\partial y}=\frac{\partial f}{\partial y} + O(h).$$
(we have used this property in the proof of the asymptotic 
expansion of the global discretization error). 

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\noindent{\bf [2]} Assume $f=f(x,y)$ is analytic, Lipschitz with respect to $y$ 
on some domain, with the Lipschtiz constant $M$, bounded and with 
bounded first-order derivatives. Assume that the exact solution
$y=y(x)$ of the initial value problem $y'(x)=f(x,y(x))$, $y(x_0)=y_0$ is
also analytic.
Show that Euler's method is convergent (you can follow some steps from the proof of the
general convergence theorem done in class, Thm. 7.2.2.3 from Stoer-Bulirsch; also you can use Lemma 7.2.2.2 done in class). The proof has to be given for this particular case.  


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\noindent{\bf [3]} Let $y_{x,h}$ be the approximate solution furnished by Euler's method
for the initial-value problem $y'=y$, $y(0)=1$. Show:

(a) $y_{x,h}=(1+h)^{x/h}$.

(b) In the expansion $y_{x,h}=e_0(x)+e_1(x)h+e_2(x)h^2+...$, with $e_0(x)=e^x$,
find $e_1(x)$ (use the fact that the exact solution is known). 

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\noindent{\bf [4]} Consider again the (IVP) 
$$y'=\frac{3x^2+4x+2}{2(y-1)}, \ \ y(0)=-1.$$

(a) Implement the classical fourth-order Runge-Kutta method and use it to 
numerically solve the (IVP) (done in the previous homework, no need to repeat). 

(b) Without using the formula of the exact solution, find an approximation 
of the error $e(x,h)=e(2,0.05)$ at step $h=0.05$ for $x=2$ (use two timesteps
$h=0.1$ and $h=0.05$, and the application from the end of section 7.2.3, 
Stoer and Bulirsch, covered in class). 
Compare this with the actual error (when the exact solution is known). 

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