Originally Posted by

**harveyo** Define $\displaystyle v(x,t) = u(x,t) - Ue (x) $

where $\displaystyle Ue (x) = x+1$

show that $\displaystyle v$ satisfies

$\displaystyle \partial v / \partial t = 2 ( \partial ^2 v / \partial x ^2)$

when

$\displaystyle 0<x<1 $

$\displaystyle t>0$

show boundary conditions on $\displaystyle v(x,t)$ are $\displaystyle v(o,t)=v(1,t)=0, t>0$

find the relevant intial condition $\displaystyle v(x,0) $ in terms of $\displaystyle f(x)$

when

$\displaystyle \partial u / \partial t = 2 ( \partial ^2 u / \partial x ^2)$

boundary conditions $\displaystyle u(0,t)=1$ $\displaystyle u(1,t)=2 , t>0$

intial condition $\displaystyle u(x,0) = f (x)$ for some function of $\displaystyle f$