Heat Equation with X> 0

Solution 1:

Hint.

Using the Laplace transform we have

$$ sU(x,s)-kU_{xx}(x,s) = 0, \ \ U_x(0,s) = G(s),\ \ \lim_{x\to \infty}U(x,s)=0 $$

Solving for $x$ we have

$$ U(x,s) = \psi_1(s)e^{\sqrt{\frac sk}x}+\psi_2(s)e^{-\sqrt{\frac sk}x} $$

but as $\lim_{x\to \infty}U(x,s)=0$ we follow with

$$ U(x,s) = \psi_2(s)e^{-\sqrt{\frac sk}x}, \ \ U_x(0,s) = G(s) $$

so

$$ \phi_2(s) = -\sqrt{\frac ks}G(s) $$

and

$$ U(x,s) = -\sqrt{\frac ks}G(s)e^{-\sqrt{\frac sk}x} $$

and finally we can use the convolution theorem

$$ u(x,t) = \mathcal{L}^{-1}\left(-\sqrt{\frac ks}e^{-\sqrt{\frac sk}x}\right)\circledast g(t) $$

NOTE

$$ \mathcal{L}^{-1}\left(-\sqrt{\frac ks}e^{-\sqrt{\frac sk}x}\right) = -\sqrt{\frac {k}{\pi}}\frac{e^{-\frac{x^2}{4k t}}}{\sqrt{t}} $$