Dynamic boundary condition for the heat equation
= Dynamic boundary condition for the heat equation
{title2=$b_t+\alpha u_x+\beta b=f$}
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.