(22) should just be the Green's function expression. I never remember these things (easier to look them up) but I think there's a typo and the exponential inside the integral should be "squared" as it were, like (26)

The formula (22) in the paper looks doable but equation (27) not so because we have to sole for both the price and the free boundary S*.

https://ijpam.eu/contents/2014-97-3/3/3.pdf

Even if we had an extra condition in (27) it would be a big numerical challenge to compute P and S*?

Certainly for European options, either Laplace in time or Mellin in stock price should just lead you to the Green's function.

(27) is a fairly well-known expression, again coming from the Green's function. From memory the name Kolodner is associated with it for heat conduction. I've got several handbooks by a couple of russians (Polyanin and another one). The books are in English. One is a handbook of solvable ODEs, an updated version of Kamke. Another is a handbook of linear PDEs, and I think they give Kolodner's formula in there

Handbook of Linear Partial Differential Equations for Engineers and Scientists 2nd Edition

(I have the older first edition)

Black-Scholes transforms into the heat conduction equation fairly easily, so you can use Kolodner's result

It's one of those problems people seem to keep doing without realizing it's been done before. People like Kim and Jacka did it in the 1990s, seemingly unaware of Kolodner, and one of the Kellers (Joe I think) did it in the 2000s seemingly unaware of either Kim/Jacka/... or Kolodner