I would like to define a 1-D heat equation using the following restrictions $0\le x\le d,\,0\le t\le 4000$ and have the outside ($x=0$) boundary defined as dependent on a function i.e. $$u=u(x,t)$$ $$u(0,t)=U(t)$$ whilst the inside boundary ($x=d$) be a Neumann boundary with zero heat transfer. Does the following notation represent that fairly? $$\left.\frac{\partial u}{\partial x}\right|_{x=d}=0$$
For background the 1D heat equation can be simplified to: $$\frac{\partial u}{\partial t}=\alpha \frac{\partial^2 u}{\partial x^2}$$ and I will be solving this PDE numerically so $U(t)$ will actually be interpolated discrete data. Thanks in advance!