Sinusoidally forced heat equation on a half-line
= Sinusoidally forced heat equation on a half-line
For $T_t=\kappa T_{xx}$ on $x>0$, with $T(x,0)=0$ and $T(0,t)=\sin\omega t$,
$$
\widetilde T(x,p)=\frac{\omega}{p^2+\omega^2}
\exp\left(-x\sqrt{\frac p\kappa}\right).
$$
Consequently the transform of $I(t)=\int_0^\infty T(x,t)\,dx$ is
$$
\widetilde I(p)=\frac{\omega\sqrt\kappa}
{\sqrt p\,(p^2+\omega^2)}.
$$