Problem computing inverse Laplace transform of the Bessel function

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.

[David Goodmanson]

Hi Kevin,

It appears that the result is just a slightly different inverse laplace transform that the toolbox can't do either. Might not exist in closed form.

[Kevin B]

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!

Answer 1

[Walter Roberson]

Maple says that for a > 0, the result is (1/2)*exp(-a/t)

[Kevin B]

Thanks very much for your help. Could you please check if it can solve the product of sum of two functions. I actually need to get the product of sums of f_s in my original post, in which case f_s will be modified to:

    f_s = 1/s*(sqrt(a1*s)*besselk(1,2*sqrt(a1*s))+sqrt(a2*s)*besselk(1,2*sqrt(a2*s)))*(sqrt(a3*s)*besselk(1,2*sqrt(a3*s))+sqrt(a4*s)*besselk(1,2*sqrt(a4*s)));

where a1, a2, a3, a4 > 0. For Matlab, I was not able to get the solution even for the single term, that's why my original post was a simplified problem.

[Walter Roberson]

Under the assumption that a1, a2, a3, a4, s are all > 0, then

(1/2)*(BesselK(1, 2*a2^(1/2)*a3^(1/2)/t)*exp((-a3-a2)/t)*a2^(1/2)*a3^(1/2)+BesselK(1, 2*a1^(1/2)*a4^(1/2)/t)*exp((-a4-a1)/t)*a4^(1/2)*a1^(1/2)+BesselK(1, 2*a2^(1/2)*a4^(1/2)/t)*exp((-a4-a2)/t)*a4^(1/2)*a2^(1/2)+BesselK(1, 2*a1^(1/2)*a3^(1/2)/t)*exp((-a3-a1)/t)*a1^(1/2)*a3^(1/2))/t

[Kevin B]

Thank you so much!

Answer 2

[BeeTiaw]

Can you look at my problem in here please? https://uk.mathworks.com/matlabcentral/answers/575788-stehfest-algorithm-for-a-problem-of-inverse-laplace-with-bessel-function