Solution
= Solution
The zero initial condition turns the time <Laplace transform> of the heat equation into
$$
p\widehat T=\kappa\widehat T_{xx}.
$$
Decay as $x\to\infty$ selects
$$
\widehat T(x,p)=\widehat f(p)e^{-x\sqrt{p/\kappa}}.
$$
The boundary value gives
$$
\widehat f(p)=a\left(\frac1p-\frac1{p+b}\right)=\frac{ab}{p(p+b)},
$$
so
$$
\boxed{\widehat T(x,p)=\frac{ab}{p(p+b)}e^{-x\sqrt{p/\kappa}}}.
$$