Write a matlab code to create a reduced model for the transient diffusion equation found in chapiter $5$ of the book on PGD

$$ Use PGD with $3$ variables: k$,$x$,$t

$$ The space$-$time$-$diffusivity domain is $\backslash$Omega x $\backslash$times $\backslash$Omega t $\backslash$times $\backslash$Omega k $=(0,1)\backslash$times $(0,0\mathrm{.}1)\backslash$times $[1,5].$

$$ For the the source term, take f $=1.$

$$ Regarding the initial and boundary conditions, take: u$($x$,$ t $=0)=$ u$($x $=0,$ t$)=$ u$($x $=1,$ t$)=0,$

$$ Use uniform grids with $61$ points for the space domain, $151$ points for the time domain and $101$ points for the diffusivity domain.

$$ Generate plots for the normalized functions Xi $($x$),$ Ti $($t$)$ and Ki $($k$)$ for i $=1,...,4.($Fig $5\mathrm{.}1,5\mathrm{.}2$ and $5\mathrm{.}3$ in the book$)$

$$ Use Finite element difference to solve the differential equations in the alternating strategy loop

$$ Make use of the $2$ matlab code provided for Poisson equation $($PGD$\_$Poisson$\_2$D$.$m and Section$\_2\_1\_3.$m$)$