I guess the question is, is what I have done mathematically correct! I believe we should take it that the formula as derived is correct given the paper is by a chap who did a PhD in the topic, has published over 50+ papers in the topic and went on to become a professor (not that professors aren't exempt from errors!). On the assumption the forumla is correct how else would/could you evaluate the integral?
I don't doubt the competence of the author and I wouldn't be able to judge it anyway.
But its hard to say if anything done here ist mathematically correct.
What you evaluated is not equivalent to what the picture you posted shows.
What the picture shows cannot be evaluated, at least not if its interpreted the way most of us obviously do by common mathematical understanding. The notation and typeseting used is furthermore not helpful. The notation with the variable as index to denote a derivation I'm used just for partial derivatives of functions in more than one variable - wouldn't have thought that theta_index_t should have that meaning. We don't know anything about Phi_index_s(x,yt) or Phi_index_0(s). Also the arbitrary use and omit of multiplication dots is not helpful, either.
As you have access to the whole document, I hope the deduction of that formula may give you further hints.
The theta_index_t notation does represent the time derivative - this is stated in the paper. I agree the omission of the multiplication "dots" is confusing.
If you have an interest I have attached the paper. Its frustrating, as without resolving this 3rd integral I am stumped solving the problem. Thanks for your time.
If you have an interest I have attached the paper.
Didn't made me any wiser on first sight. Especially the step from equ (20) to (21) is unclear to me. should be "just" the time derivative and a lim x-->0 and sign change.
With some perseverence I have been able to solve the problem via another route. The same author pubilshed another paper that derivess the same problem equations using a slightly different method where he provides final equations without the mixed integral with Laplace variable "s" and time functions in the final integral.
The final equation is attached as pdf.
I have used the equations provided to benchmark some graphs in his paper and get the same results! The attached sheets show the solution and divergent unstable behaviour is predicted.
Many thanks for your assistance and interest in the problem. Have a good Xmas.