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\noindent{\bf HW \#1. Math 269C (due on Friday, April 14)}
\noindent{\bf [1]} Show that if $w$ is continuous on $[0,1]$, and
$$\int_{0}^1wvdx=0,\mbox{ for all }v\in V,$$
with
$V=\{v:[0,1]\rightarrow R,\ continuous,\ v(0)=v(1)=0,\ v' \ piecewise-continuous\ and\ bounded\}$,
then $w(x)=0$ for $x\in[0,1]$.
\noindent{\bf [2]} Construct a finite-dimensional subspace $V_h$ of $V$ (from problem [1])
consisting of
functions which are quadratic on each subinterval $I_j$ of a partition of $I=(0,1)$. How can one choose the parameters to describe such functions ? Find the
corresponding basis functions. Then formulate a finite element method for
$(D)$ using the space $V_h$ and write down the corresponding linear system of
equations in the case of a uniform partition.
Recall that $(D)$ is
$$-u''=f\mbox{ in }(0,1),\ \ u(0)=u(1)=0.$$
\noindent{\bf [3]} Consider the BVP
$$\frac{d^4u}{dx^4}=f,\ \ 0