Along the whole positive $x$ -axis, we have an ideal heat-conducting rod, the surface of which is isolated. The initial temperature of the rod is 0 degrees. Determine the temperature function $u(x,\,t)$ when at the time $t = 0$

(a) the head $x = 0$ of the rod is set permanently to the constant temperature;

(b) through the head $x = 0$ one directs a constant heat flux.

The heat equation in one dimension reads

For (a), the second boundary condition implies $\displaystyle U(0,\,s) = \frac{u_0}{s}$ . But by (2) we must have $U(0,\,s) = C_2\!\cdot\!1$ , whence we infer that $\displaystyle C_2 = \frac{u_0}{s}$ . Accordingly, $$U(x,\,s) \;=\; u_0\cdot\frac{1}{s}e^{-\frac{x}{c}\sqrt{s}},$$ which corresponds to the solution function $$u(x,\,t) \;:=\; u_0\,\mbox{erfc}\frac{x}{2c\sqrt{t}}$$ of the heat equation (1).

For (b), the second boundary condition says that $\displaystyle U_x'(0,\,s) = -\frac{k}{s}$ , and since (2) implies that $U_x'(x,\,s) =-\frac{\sqrt{s}}{c}C_2e^{-\frac{\sqrt{s}}{c}x}$ , we can infer that now $$C_2 \;=\; \frac{ck}{s\sqrt{s}}.$$ Thus $$U(x,\,s) \;=\; \frac{ck}{s\sqrt{s}}e^{-\frac{x}{c}\sqrt{s}},$$ which corresponds to $$u(x,\,t) \;:=\; k\left[2c\sqrt{\frac{t}{\pi}}e^{-\frac{x^2}{4c^2t}}-x\,\mbox{erfc}\frac{x}{2c\sqrt{t}}\right]\!.$$

[Not ready . . .]