Does anyone know what is the solution of the heat equation
$\displaystyle u_t(x,t)=u_{xx}(x,t)$
with mixed boundary conditions
$\displaystyle u_x(0,t)=0$
$\displaystyle u_x(1,t)=\epsilon u(1,t)$
At x=0 there is no flow of temperature.
At x=1 there is flow of temperature.
But there must be initial conditions: the initial distribution of temperature
u(x,0)=f(x) which changes in time due to flow of temperature (boundary conditions).
You may try to solve by separation of variables
u(x,t)=p(x)q(t).