I need to invert a Laplace transform to obtain the temperature variation in a particular problem in time and (1-D) space. In the frequency domain, I have
$$F(s)=\frac{1}{s}e^{H(s)\xi}$$
where
$$H(s)=\frac{1}{C}\Big[s + Q\sqrt(s)\frac{K_1(\sqrt{s})}{K_0(\sqrt{s})}\Bigr]$$
and where $K_0()$ and $K_1()$ are the Modified Bessel Functions of the Second Kind, of orders $0$ and $1$, respectively. The function $H(s)$ has a branch point at $s=0$, but a closed contour can be generated with a branch cut as shown in the figure below
Noting that the integrand of the Bromwich integral vanishes on the large semicircle as its radius tends to $\infty$, and with the use of the Cauchy Integral Theorem, the temperature distribution is (skipping details) given by the integral in real space:
$$\theta(\tau,\xi) = 1-\frac{2}{\pi} \int_0^\infty \frac{1}{u} e^ {-\Bigl[ u^2\Big(\tau-\frac{\xi}{C}\Big) +\frac{Q}{C}uA(u)\xi\Bigr]} \sin\Big(\frac{Q}{C}uB(u)\xi\Big) \text{d}\xi$$
where $\tau$ and $\xi$ are the dimensionless temporal and spatial coordinates in the physical domain, and where $A(u)$ and $B(u)$ are functions involving the Modified Bessel functions. The term $C$ can be thought of as the wave speed in the underlying first order PDE.
For large times, the solution matches that obtained from a Gaver-Stehfest function-sampling algorithm, although I do not necessarily state this as proof of correctness.
The problem however is that for smaller times, when $\xi/C > \tau$, the solution breaks down since the exponential argument becomes unbounded as $u\rightarrow \infty$ which is evident from the solution above. At small times $s\rightarrow\infty$ and the ratio of the Modified Bessel functions tends to unity in the expression for $H(s)$. But this will still not resolve the problem caused by the condition $\xi/C > \tau$. Is it the case that the solution is only expected to be valid for $\tau > \xi/C >$? It appears to be that the spatial location cannot be ahead of the distance traversed by the thermal front at the wave speed in the elapsed time.
I am really interested in the short term behaviour, so it's a pity if this is the case, after all of the hard work that I put in working out the contour integral! Any suggestions therefore for how to work around this issue would be greatly appreciated.
I believe that I may have answered my own question within it, and that the solution is indeed valid only for locations that are upstream of and at the thermal front propagating at its wave speed. For all downstream locations, the solution remains at the initial condition.