Inverse Laplace Transform of $e^{-x\sqrt{s}}/\left[\left(s-a^2\right)\left(\sqrt{s}+a\right)\right]$

Question

I'm interested in the following Laplace inversion problem $$f(t,x)=\mathcal{L}^{-1}\left\{\frac{e^{-x\sqrt{s}}}{\left(s-a^2\right)\left(\sqrt{s}+a\right)}\right\}$$ with $a$ real and positive, as appears in Eq. (2) of Phys. Rev. E 96, 023104 (a pdf can be found here). In that article, the authors mention the following early-time approximation $$f(t,x)=2\sqrt{\frac{t}{\pi}}e^{-x^2/4t}-x{\rm Erfc}\left(\frac{x}{2\sqrt{t}}\right),$$ but do not solve the full Laplace inversion problem. Any suggestions on how to solve this would be much appreciated.

Answer 1

With CAS like: Maple 2022 I have:

$$\mathcal{L}^{-1}\left\{\frac{e^{-x\sqrt{s}}}{\left(s-a^2\right)\left(\sqrt{s}+a\right)}\right\}=\\\sqrt{\frac{t}{\pi}}\, {\mathrm e}^{-\frac{x^{2}}{4 t}}+\frac{-\mathrm{erfc}\! \left(\frac{2 a t +x}{2 \sqrt{t}}\right) {\mathrm e}^{a \left(a t +x \right)} \left(4 a^{2} t +2 a x +1\right)+\mathrm{erfc}\! \left(-\frac{2 a t -x}{2 \sqrt{t}}\right) {\mathrm e}^{a \left(a t -x \right)}}{4 a} $$ for $x>0$.

Maple code:

(t/Pi)^(1/2)*exp(-1/4*x^2/t)+1/4*(-erfc(1/2*(2*a*t+x)/t^(1/2))*exp(a*(a*t+x))*( 4*a^2*t+2*a*x+1)+erfc(-1/2*(2*a*t-x)/t^(1/2))*exp(a*(a*t-x)))/a

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