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 $g(x,t)$ ?
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 $X(0)=2\,X(1)$, $X'(0)=X'(1)$.