Inverse Laplace transform of exponential form

Question

As homework form my turbulence class, I have to solve the heat equation on a semi-infinite domain where the fixed boundary condition is a function that depends in time. I decided to use Laplace transforms to find the solution. After transforming and evaluating the Boundary condition I get this solution in the "Laplace space":

$$\hat{c}(x,s) = \frac{as}{s^2 + \omega^2}e^{-\sqrt{\frac{s}{\kappa}}x} + \frac{T_0}{s}.$$

So far I looked in tables but I cannot find an inverse transform for the term

$$\frac{s}{s^2 + \omega^2}e^{-\sqrt{\frac{s}{\kappa}}x}.$$

I tried to do the transformation using the convolution of the functions

$$F(s) = \frac{s}{s^2 + \omega^2} \hspace{13pt} \text{and} \hspace{13pt} G(s)=e^{-\sqrt{\frac{s}{\kappa}}x},$$

which have a defined inverse transform, but the final integral is too difficult to solve. I also tried using partial fractions but I doesn't work.

Anybody has an idea or has done something similar in the past?

This is the problem that I'm solving:

Solve the heat equation for a semi-infinite medium (at rest) limited by a surface submitted to an oscillating temperature $T = T_0 + a\cos(ωt)$.

Answer 1

With CAS like Mathematica I have:

$$\mathcal{L}_s^{-1}\left[\frac{s \exp \left(-\sqrt{\frac{s}{\kappa }} x\right)}{s^2+\omega ^2}\right](t)=\frac{1}{4} e^{-\frac{x \left(\sqrt{-i \omega }+\sqrt{i \omega }\right)}{\sqrt{\kappa }}-i t \omega } \left(e^{x \sqrt{\frac{i \omega }{\kappa }}} \text{erfc}\left(\frac{x-2 t \sqrt{-i \kappa \omega }}{2 \sqrt{t \kappa }}\right)+e^{\frac{x \left(2 \sqrt{-i \omega }+\sqrt{i \omega }\right)}{\sqrt{\kappa }}} \text{erfc}\left(\frac{x+2 t \sqrt{-i \kappa \omega }}{2 \sqrt{t \kappa }}\right)+e^{2 i t \omega } \left(e^{x \sqrt{-\frac{i \omega }{\kappa }}} \text{erfc}\left(\frac{x-2 t \sqrt{i \kappa \omega }}{2 \sqrt{t \kappa }}\right)+e^{\frac{x \left(\sqrt{-i \omega }+2 \sqrt{i \omega }\right)}{\sqrt{\kappa }}} \text{erfc}\left(\frac{x+2 t \sqrt{i \kappa \omega }}{2 \sqrt{t \kappa }}\right)\right)\right)$$

Mathematica code:

InverseLaplaceTransform[(s/(s^2 + \[Omega]^2))*Exp[(-Sqrt[s/\[Kappa]])*x], s, t] == (1/4)*E^(-((x*(Sqrt[(-I)*\[Omega]] + Sqrt[I*\[Omega]]))/Sqrt[\[Kappa]]) - I*t*\[Omega])* (E^(x*Sqrt[(I*\[Omega])/\[Kappa]])*Erfc[(x - 2*t*Sqrt[(-I)*\[Kappa]*\[Omega]])/(2*Sqrt[t*\[Kappa]])] + E^((x*(2*Sqrt[(-I)*\[Omega]] + Sqrt[I*\[Omega]]))/Sqrt[\[Kappa]])* Erfc[(x + 2*t*Sqrt[(-I)*\[Kappa]*\[Omega]])/(2*Sqrt[t*\[Kappa]])] + E^(2*I*t*\[Omega])*(E^(x*Sqrt[-((I*\[Omega])/\[Kappa])])*Erfc[(x - 2*t*Sqrt[I*\[Kappa]*\[Omega]])/(2*Sqrt[t*\[Kappa]])] + E^((x*(Sqrt[(-I)*\[Omega]] + 2*Sqrt[I*\[Omega]]))/Sqrt[\[Kappa]])*Erfc[(x + 2*t*Sqrt[I*\[Kappa]*\[Omega]])/(2*Sqrt[t*\[Kappa]])]))