One$-$Dimensional Heat Equation

Using MATLAB: Consider a slab with heat applied to the left and$/$or right side. As heat is applied, the temperature, $T,$ at any one location

in the slab will increase. Thus, $T$ is a function of time. Additionally, there will be a temperature gradient along the slab.

Thus, $T$ is a function of space.

The PDE defining the temperature $T(x,t)$ across this slab at any time $t$ and position $LxR$ is given by

$\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}-\frac{C}{n}*\frac{de{l}^{2}T}{del{x}^2}=0$

where $n=2\frac{J}{g}K$ is the specific heat, $=2\mathrm{.}7{10}^4\frac{g}{m^3}$ is the material density, and $C=\frac{{205}^W}{m}K$ is the thermal

conductivity. Assume a length of the slab is $5m.$ Assume time ranges from $0$ to $100$ seconds.

PART $1$

Assume boundary conditions of $T_L(t)=50+t$ and $T_R(t)=95+t.$ Solve and surf plot the temperature throughout

the slab for initial condition of $T_0(x)=0$ in position $1$ of a $12$ figure. $(2$ pts$)$

Re$-$run $1)$ but plot in position $2$ of the $12$ figure assuming $T_0(x)=20**|x-2\mathrm{.}5|-65.(1$ pt$)$

In general, where do you see the effect of the initial condition and what effect does it have? Verify that the initial

and boundary conditions are what you expect them to be$.(2$ pts$)$

PART $2$

Assume the system changes slightly. Now, it has an initial condition of $T_0(x)=100$ and boundary conditions of

$T_L(t)+0\mathrm{.}35\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}(L,t)=0$ and $T_R(t)-0\mathrm{.}35\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}(R,t)=0.$ Solve and surf plot the temperature for a $10$ second

simulation throughout the slab in position $1$ of a $12$ figure $($Figure $2).(2$ pts$)$

Re$-$run $4)$ but plot in position $2$ of the $12$ figure assuming $qR=0\mathrm{.}15.$ Why does temperature rise over time,

symmetrically about $x$ in $4),$ but have a very different trend with $qR$ is changed? What does this tell you about flux

at boundaries with specific respect to directionality? $(3$ pts$)$