# Problem computing inverse Laplace transform of the Bessel function

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Kevin B on 11 Nov 2017
Answered: BeeTiaw on 5 Aug 2020
I am trying to evaluate the inverse Laplace transform of a function that includes the modified Bessel function of the second kind, i.e., my function is f_s = sqrt(a*s)*besselk(1,2*sqrt(a*s)), where a is a constant (say a = 0.5) and 's' is defined as a symbol. To evaluate the inverse Laplace transform of this function, I used the built-in Matlab function 'ilaplace' as transV = ilaplace(f_s,s,t), where I want to evaluate my function at 't' = 2. However, the problem I am getting is that the inverse transformed expression transV still contains 's' variable, which I supposed to transform to 't' after the inverse Laplace transform.
Can anyone help me with what is the problem with this approach? I checked the Matlab documentation and still couldn't figure out the solution. I already tried using 'vpa' function as it was suggested in one of the Matlab forums, but that didn't help to get rid of 's'. I am stuck with this and would really appreciate any help on it.
My code is
a = 0.5; t = 2;
syms s;
f_s = 1/s*sqrt(a*s)*besselk(1,2*sqrt(a*s));
transV = ilaplace(f_s,t); % or transV = vpa(ilaplace(f_s,t));
With this, the final result transV still has an expression which is a function of 's' although it should be transformed after ilaplace.
##### 2 CommentsShowHide 1 older comment
Kevin B on 11 Nov 2017
Hi David,
Thanks for your suggestion. Yes, the closed form solution seems not to exist. I tried Mathematica to check if it can help but without success. For me, the numerical solution is fine for now and I am not able to get that either. Do you have any further suggestions to get the numerical solution? Thanks a ton!

Walter Roberson on 11 Nov 2017
Maple says that for a > 0, the result is (1/2)*exp(-a/t)
Kevin B on 12 Nov 2017
Thank you so much!

BeeTiaw on 5 Aug 2020

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