# No flux Neumann boundary condition for non-stationary PDE equivalent to Dirichlet boundary?

When using no flux Neumann boundary conditions (i.e. zero derivative to the normal on the boundary) in a non-stationary PDE, I don't seem to recognize the difference to using Dirichlet boundary conditions directly...

Since the initial values prescribe the values that the boundary has to take at $$T = 0$$ and we impose that these values must not change through no flux, wouldn't this be equivalent to simply prescribing the initial values at all times $$T$$ at the boundary (i.e. Dirichlet boundary, constant w.r.t. time)?

$$\frac{dT}{dt} - \nabla^2 T = S$$
If $$S \neq 0$$, then you have a source term that changes the value of the temperature as time evolves. Consequently, even if your initial condition is $$T(\textbf{x},t)=0$$, your temperature will increase because of the source term. Then, having a Dirichlet or a Neumann boundary condition will lead to completely different result. If you have a Dirichlet (say $$T=0$$) then that boundary will remain at 0. If you have a Neumann, the gradient at the boundary will remain 0, thus the temperature will increase due to the source term. Consequently, you would get completely different temperature profile in the end.