4 7.24, §4 Solve

Solve the PDE:

$$s \hat u = \kappa \hat u_{xx}$$

This is a constant coefficient ODE in $x$, with $s$ simply a parameter. Solve from the characteristic equation:

$$s = \kappa k^2 \quad\Longrightarrow\quad k = \pm \sqrt{s/\kappa}$$

$$\hat u = A e^{\sqrt{s/\kappa} x} + B e^{-\sqrt{s/\kappa} x}$$

Apply the BC at $x=\infty$ that $u$ must be regular there:

$$A=0$$

Apply the given BC at $x=0$:

$$\hat u_x = \hat g(s) \quad\Longrightarrow\quad - B \sqrt{\frac{s}{\kappa}} = \hat g$$

Solving for $B$ and plugging it into the solution of the ODE, $\hat u$ has been found:

$$\hat u = - \sqrt{\frac{\kappa}{s}} e^{-\sqrt{s/\kappa} x} \hat g$$