Consider the one$-$dimensional heat transfer problem seen in class with the same boundary conditions.

In this case, we need to discretize our problem into $4$ elements of equal length. Thus, $5$ points will define the temperature for the entire domain $($x$1,$ x$2,$ x$3,$ x$4,$ x$5)$ with x$5=$L$.$ Consider the same shape functions seen in class such that $$i$($xi$)=1$ and $$i$($xi$+1)=0$ and vice versa $$i$+1($xi$)=0$ and $$i$+1($xi$+1)=1.$

$1.-$ Obtain the approximation of the temperature in each element using the finite element method by Galerkin.

a$)$ Obtain the assembled matrix of the elements

b$)$ Find the solution to the variables ai$,$ i$=\{1,...,5\}$