During solving convection-diffusion equation using Laplace transformation, I obtained the following as a part of the solution in Laplace time-domain. I wonder if it can be transformed into the real-time domain. $$\mathcal{L}^{-1}\left\{\frac1{s}\ e^{ -{ a \sqrt{b + c\ (d \sqrt{s} + s})}} \right\}$$
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Only with infinite double sum: $$\mathcal{L}_s^{-1}\left\frac{\exp \left(-a \sqrt{b+c \left(d \sqrt{s}+s\right)}\right)}{s}\right=\sum _{n=0}^{\infty } \sum _{j=0}^{\infty } \frac{\left(-\frac{1}{2}\right)^j a^j \left(\frac{1}{c}\right)^{-\frac{j}{2}+n} c^n d^n \sqrt{\pi } \left(\frac{1}{t}\right)^{\frac{1}{2} (j-2 n)} t^{-\frac{n}{2}} , _1F_1\left(-\frac{j}{2}+n;\frac{1}{2} (2-j+n);-\frac{b t}{c}\right)}{\Gamma \left(\frac{1}{2}+\frac{j}{2}\right) \Gamma \left(1+\frac{j}{2}-n\right) \Gamma \left(1-\frac{j}{2}+\frac{n}{2}\right) \Gamma (1+n)}$$ Probably closed-form not exist. – Mariusz Iwaniuk Aug 15 '19 at 09:14