 Dynamic boundary condition for the heat equation

A dynamic boundary condition includes a time derivative of the boundary trace. For a half-line heat equation, $u_t(0,t)+\alpha u_x(0,t)+\beta u(0,t)=f(t)$ gives a separate boundary evolution coupled to the interior flux. Its Laplace transform contains the initial trace $u_0(0)$. The sign of $u_x$ differs from that of the outward normal derivative at a left endpoint.