now by defining several dimensionless variables, we got another form of diffusion equation:

$\displaystyle U_{T}=U_{XX} \ \ \ \ T>0, \ \ 0<X<1 $

X,T,U here are all dimensionless.

U with initial conditions:

$\displaystyle U(X,0)=0, \ \ 0\leq X\leq1 $ and boundary conditions :

$\displaystyle U(0,T)=T, \ \ U(1,T)=0, \ \ T\geq0 $

$\displaystyle U(X,T)=T(1-X)+V(X,T) $

find the analytic solution of this problem.