Problem $1$

Consider the steady state heat diffusion equation in a rod of extending from $x=0$ to $x=L$ :

$\frac{d^{2}T}{(d)x^2}=-k{e}^{-\frac{(x-a{)}^2}{b^2}}.$

The right$-$hand$-$side of this equation is a source term that contributes to heating the rod and is a function

of the $x$ location on the rod where $k,a,$ and $b$ are constants.

$(10$ pts$)$ Using central differencing, discretize equation $(1)$ at an arbitrary interior point $x_{i}N=5T=atx=0$

$T=atx=LN=100k=1000,a=0\mathrm{.}5,b=0\mathrm{.}1x=0x=1,=300K=350KN=5T=atx=0$

$\frac{dT}{(d)x}=atx=LN=100k=1000,a=0\mathrm{.}5,b=0\mathrm{.}1x=0x=1,=300K=-10\frac{K}{m}i(1.$

$Do$ this symbolically and use $x_{i}in$ the source term

$(10pts)$ For $N=5$ grid points, show the system $of$ equations $(in$ matrix form$)$ that must $be$ solved

subject $to$ the Dirichlet boundary conditions:

$T=atx=0$

$T=atx=L].$

Please asnwer all, will upvote