In solutions to the heat equation $u_t(x,t)=cu_{xx}(x,t)$ I've seen they've used the set of boundary conditions
$$u(0,t)=u(L,t)=0$$ $$u(x,0)=u_0(x)$$
These set of boundary conditions is set to model a situation in which a rod with some initial heat distribution is submerged in 0 degrees Celsius. I'm interested in using the heat equation to model the diffusion of particles in a closed system, and therefore my system must conserve mass.
In other words, I need to solve
$$u_t(x,t)=cu_{xx}(x,t)$$ $$u(x,0)=u_0(x)$$ $$ \int_0^L u(x,t) \, dx = \text{Constant}$$
I know there are many posts on this site that deal with solving the heat equations, but I haven't found any that deal with this specific set of boundary conditions. Can someone point me in the directions of solving this equation?
If an end is isolated, the $x$ derivative there must be set to zero: $$u_x(0,t)=u_x(L,t)=0\tag{1}$$ This will imply the conservation of mass: $$ \frac{d}{dt}\int_0^L u(x,t) \, dx =\int_0^L cu_{xx}(x,t) \, dx =c (u_x(L,t)-u_x(0,t))=0 $$
The physical meaning of $(1)$ becomes transparent if we recall that the flow of heat is proportional to the gradient: it's $-cu_x$. So, zero flow of heat through an endpoint means $u_x$ being zero.