1
$\begingroup$

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 $T = T_0 + a\cos(ωt)$.

$\endgroup$
0
$\begingroup$

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]])]))

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.