= Neumann boundary-forcing formula
{c}
For $u_t=u_{xx}+\beta u_x$ with $u_x(0,t)=g(t)$ and $u_x(L,t)=h(t)$, two applications of <integration by parts> against a <Neumann eigenfunction> give the boundary forcing $e^{\beta L}\phi_k(L)h-\phi_k(0)g$. Solving the resulting modal <linear differential equation> gives
$$
u(x,t)=\int_0^LK_\beta(x,y,t)e^{\beta y}u_0(y)\,dy+\int_0^t\left[e^{\beta L}K_\beta(x,L,t-s)h(s)-K_\beta(x,0,t-s)g(s)\right]ds.
$$
The endpoint <derivatives> are interior limits: the short-time singularity prevents evaluating each homogeneous boundary derivative before the time integral and infinite sum. The constant <eigenfunction> gives the weighted-mass law $\partial_t\int_0^Le^{\beta x}u\,dx=e^{\beta L}h-g$.
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