I'm a bit confused by the difference between a Neumann condition and a Robin condition. In the textbook a Neumann condition is defined as a normal derivative that is specified on the boundary. It's in the form of $\forall t, \partial u/\partial \textbf{n}=g(\textbf{x},t)$ for some function $g$ where $\textbf{x}$ is on the boundary of domain $D$. A Robin condition is in the form of $\forall t, \partial u/\partial \textbf{n}+au=g(\textbf{x},t)$ for some function $g$. Why can't I move the $au$ term in the Robin condition to the right hand side and make it a Neumann condition? What's the point of classifying them into two types of conditions?
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It might be useful to think, in both of these equations, of $g$ as the known/given function and $u$ as the unknown function one is trying to solve the equation for. Accordingly the formatting is so that all terms involving the unknown function are on the left handside and the terms involving the known function(s) are on the right handside.
Another way to think of the difference is that the Neumann boundary conditions involve an operator that only contains a first order normal ("outward") derivative, where as the Robin boundary conditions involve an operator that contains both a first order and a zeroth order normal derivative (and the Dirichlet boundary conditions involve an operator that contains only a zeroth order normal derivative).