Fixed-temperature and fixed-flux heat equation on an interval ($u_t=\nu u_{xx},u(0,t)=0,k{u}_x(L,t)=H$)

For positive thermal diffusivity $\nu$, thermal conductivity $k$ and length $L$, subtract the stationary solution $w(x)=Hx/k$. The remainder satisfies the heat equation with a homogeneous Dirichlet boundary condition at zero and a homogeneous Neumann boundary condition at $L$. The mixed Dirichlet-Neumann modes on an interval are $\sin (k_{n}x)$, where $k_n=(n+1/2)\pi /L$. Thus for an initial temperature profile $u_0$,For zero initial temperature, $b_n=-2H(-1{)}^n/(kL{k}_n^2)$. The series gives the initial profile in the usual Fourier series sense and a smooth solution for $t>0$. If $H\ne 0$, the zero initial profile does not satisfy the prescribed endpoint derivative at $t=0$: the temperature can be continuous there without its spatial derivative extending continuously to that corner. The sign $k{u}_x(L,t)=H>0$ describes incoming heat flux, since the outward conductive heat flux is $-k{u}_x$.