Laplace transform method for the heat equation

Example

Consider the heat equation on the half-line, for an unknown function $u(x, t)$ with $x > 0$ and $t > 0$: $$u_t = u_{xx}, \qquad u(x, 0) = 0, \qquad u(0, t) = 1, \qquad u \text{ bounded}.$$ This models a cold rod, initially at temperature $0$, whose end $x = 0$ is held at temperature $1$ from time $0$ on. Let $U(x, s) = \int_0^\infty u(x, t) e^{-s t} \, dt$ be the Laplace transform in the time variable, for real $s > 0$. Since $u(x, 0) = 0$, the transform of $u_t$ is $s U$, so the equation becomes the ordinary differential equation $U_{xx} = s U$ in $x$. Its solutions are $U = A(s) e^{-x \sqrt{s}} + B(s) e^{x \sqrt{s}}$. Boundedness in $x$ forces $B(s) = 0$, and the boundary condition gives $U(0, s) = \mathcal{L}\{1\}(s) = 1/s$, so $A(s) = 1/s$. Hence $U(x, s) = e^{-x \sqrt{s}} / s$. The table pair $\mathcal{L}\{\operatorname{erfc}(x / (2 \sqrt{t}))\}(s) = e^{-x \sqrt{s}} / s$, for fixed $x > 0$, gives the inverse transform: $$u(x, t) = \operatorname{erfc}\left(\frac{x}{2 \sqrt{t}}\right),$$ where $\operatorname{erfc}$ is the complementary error function. Check: as $t \to 0^+$ with $x > 0$ fixed, $\operatorname{erfc}(x/(2\sqrt{t})) \to \operatorname{erfc}(+\infty) = 0$; at $x = 0$, $\operatorname{erfc}(0) = 1$; and $0 \le u \le 1$ for $x \ge 0$, so $u$ is bounded.