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I have the heat equation on a finite interval with these periodic-like boundary conditions,

$$\left\{ \begin{matrix}u_t=u_{xx}, \qquad \qquad \qquad 0<x<1, t>0\\ u(0,t)={\bf2}u(1,t), \qquad \qquad \qquad \quad t>0\\ u_x(0,t)=u_x(1,t), \qquad \qquad \qquad \quad t>0\\ u(x,0)=f(x),\qquad \qquad\qquad 0<x<1\end{matrix}\right.$$

Do you know of a suitable transformation $F$ to the function $u(x,t)$ of the form $g(x,t)=F[u(x,t)]$ that converts the boundary conditions of $u(x,t)$ to one of the common (Dirichlet, Neumman, Robin, Mixed, Periodic) on $g(x,t)$ ?

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  • $\begingroup$ Periodic conditions can result in two-dimensional eigenspaces, whereas separated conditions result in one-dimensional eigenspaces. So these are fundamentally different in general. $\endgroup$ – DisintegratingByParts Feb 28 '17 at 13:46
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No, as far as I know there is not such a transformation. However the equation can be solved by the usual way of separating variables, that is, looking for solutions of the form $u(x,t)=X(x)\,T(t)$. This will lead to the eigenvalue problem $X''=\lambda\,X$ with $X(0)=2\,X(1)$, $X'(0)=X'(1)$.

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  • $\begingroup$ I was thinking that a transformation to other boundary conditions would be like a "tricky" solution, but I couldn't find any. Another quick question, do you know how to generalize periodic boundary conditions to higher dimensions ? , or this periodic-like boundary condition, I believe the generalization should be similar. I can see it for example in a rectangular domain in $ \mathbb{R}^2$, but in other domains is not clear for me $\endgroup$ – Keith Feb 25 '17 at 21:19

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