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I have solved thermal energy balance equation during cold fluid injection into a semi-infinite model. I obtained the following solution in Laplace domain.

$$\bar T_D = \frac{r_D^vK_v(\sqrt[]s r_D)}{sK_v(\sqrt[]s)}$$

How can I obtain inverse Laplace transformation of this solution given $K_v$ is modified Bessel function of second kind and order $v$. $v $ is an integer number higher than zero.

Refaat Galal
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  • What is the specific function $K$? – Matthew Cassell Oct 03 '19 at 18:41
  • Kv is Modified Bessel function of second kind and integer order v. @Mattos – Refaat Galal Oct 03 '19 at 18:42
  • Most likely you'll have to do the complex line integral definition of the ILT. – Adrian Keister Oct 03 '19 at 18:54
  • I hope anyone can help me obtain this inversion. I am sorry but as a petroleum engineer, my mathematical skills are not high.@AdrianKeister Keister – Refaat Galal Oct 03 '19 at 19:02
  • It seems someone has already tried something very similar here. – Matthew Cassell Oct 04 '19 at 15:07
  • Yes. I have checked his solution before posting my question. He has solved diffusion equation, so his solution includes modified Bessel functions of zeroth order. In my case, my solution is for convection-diffusion problem, so the solution contains modified Bessel functions of order "Beta". "Beta" physically refers to Peclet number (Heat transfer by convection / Heat by diffusion). If "Beta" is set equals to zero, convection diffusion equation is simplified to diffusion equation, and thereby my solution is simplified to the solution that you referred to.@Mattos – Refaat Galal Oct 04 '19 at 17:30
  • Can you please help me obtain the inversion in my post?@Ron Gordon – Refaat Galal Oct 04 '19 at 18:15

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