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Given the pde ## \displaystyle k\frac{\partial^2 T}{\partial x^2}=\rho c_0 \frac{\partial T}{\partial t}## and the BC

##T(0,t)=T_\infty## and ##T(L,t)=T_\infty## for ##t>0## and the initial condition ##T(x,0)=T_0##

The author proceeds to 'normalize' the PDE in order to make the BC homogeneous. He has the following

## \displaystyle \alpha=\frac {k}{\rho c_0}##, ## \bar x = x/L##, ##\displaystyle \bar t = \frac {\alpha t}{L^2}##, ##\displaystyle u=\frac{T-T_\infty}{T_0-T_\infty}##

This leads to

##\displaystyle -\frac{\partial^2 u}{\partial x^2}+ \frac{\partial u}{\partial t}=0##

##u(0,t)=0##, ##u(1,t)=0## and ##u(x,0)=1##

1)How did he arrive at the first and third line from the bottom?

2) Why does he make the BC's homogeneous?

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# Normalizing the PDE to make BC's homogeous

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