= Solution
For a nonzero separated solution $v=X(x)T(t)$, <separation of variables> gives $X''+\kappa^2X=0$ and $T'=-\nu\kappa^2T$. The endpoint conditions require $X=\sin\kappa x$ and $\cos\kappa L=0$. Thus the <mixed Dirichlet-Neumann modes on an interval> have $\kappa_n=(n+1/2)\pi/L$, $n\geq0$. Their squared <L2 norms> are $L/2$, so the initial <Fourier coefficients> are
$$
b_n=-\frac{2H}{kL}\int_0^Lx\sin(\kappa_nx)dx=-\frac{2H(-1)^n}{kL\kappa_n^2}.
$$
The integration uses $\cos\kappa_nL=0$ and $\sin\kappa_nL=(-1)^n$. Therefore
$$
\boxed{v(x,t)=-\frac{8HL}{k\pi^2}\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^2}\sin\frac{(2n+1)\pi x}{2L}\exp\left[-\frac{\nu(2n+1)^2\pi^2t}{4L^2}\right],\qquad u=\frac{Hx}k+v.}
$$
At positive time the series can be differentiated termwise and satisfies both homogeneous endpoint conditions for $v$. At time zero it recovers $-Hx/k$, so $u$ recovers the initial zero <temperature>. If $H\ne0$, the prescribed <derivative> at $(L,0)$ is incompatible with differentiating the initial zero profile; the <temperature> is still continuous there, while its spatial <derivative> has a corner layer. At long times the decaying modes leave $u\to Hx/k$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1-heated-bar-mixed-boundaries.png]
{title=Heating a bar through one end while the other end stays at zero temperature}
{description=Original dimensionless temperature profiles at several diffusion times. The eventual linear profile has the imposed unit derivative at the heated end.}
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