The diffusion equation for heat expressed in terms of temperature and with uniform influx of heat

at a rate of the power $p$ is

$\frac{\text{d}\text{e}\text{l}\text{T}(x,t)}{\text{d}\text{e}\text{l}\text{t}}=\frac{}{c_p}\frac{de{l}^{2}T(x,t)}{del{x}^2}+\frac{p}{c_p}.$

Here, $$ is the thermal conductivity, $$ the density, and $c_p$ the heat capacity. The term $\frac{p}{c_p}$

accounts for a uniform and constant influx of heat with power $p($heat $q$ added per unit time,

del$\frac{q}{d}$elt. Assume a one dimensional sample of length $2d(-dxd)$ with constant temperature $T_0$

at both end$-$points and steady state conditions, i$.$e$.,$

$\frac{\text{d}\text{e}\text{l}\text{T}(x,t)}{\text{d}\text{e}\text{l}\text{t}}=0.$

a$)$ Treating $,,c_p,$ and $p$ as constants, determine the steady state $($time independent$)$

temperature profile $T(x)$ with the boundary condition that $T(x=-d)=T(x=d)=T_0,$ leading

to $T>T_0$ for all $80p.T(x)T-dxd-d,$ see sketch below.

$80p.$

$b$