Problem $1(5$ points, CCOs #$5,10)/$ Gradescope

The temperature of a fluid in a pipe of length $L$ can be described by the following $1$D non$-$dimensional equation

$\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}=-u\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{x}}+(T)\frac{de{l}^{2}T}{del{x}^2}+Q(x,t)$

with time $t,$ temperature $T,$ constant velocity $u,$ temperature dependent thermal diffusivity $(T),$ and a source term describing additional heating$/$cooling along the pipe, $Q(x,t).$ On the left end, the fluid entering the pipe has a prescribed temperature of