consider the heat equation
$u_t=a(t)u_{xx}+f(x,t)$, $0<x<L$, $0<t<T$
subject to the initial condition
$u(x,0)=g(x)$
and boundary conditions
$u(1,t)=0,$ $u_x(0,t)+hu(0,t)=0$
where the functions $f,g$ and the constant$h$ is known. My question is about meaning of this model in diverse scientific field. Such as in physics, chemistry, biology etc. can you give me some example about what it models in these area ? What does the boundary conditions model? and Is there any relationship between $a(t)$ and $h$?
Please help me about this problem. I really wonder the answers of these questions.
Thanks in advance!
For example, it can model a process of heating a rod with time-depending heat conductivity coefficient (it can either rise or decay), for example, there is a chemical reaction in a rod, and a material (consequently, conductivity coefficient) changes with time. The boundary condition tells us that on one side of rod the temperature is constant, and on another a heat flow is constant (you might have made a mistake in a text, because in this moment your problem is ill-posed), which says us, for example, that there is a constant heat exchange with something. Here $a$ and $h$ can be both dependent and independent, because $a $ is a characteristic of rod, and $h$ is a characteristic of media, in which this rod is situated