Good day community, I try to solve the 2D heat equation in cylindrical coordinates. I wanna follow a paper (Selim et al.: Temperature rise in a semi-infinite medium heated by a disc source) and therefortherefore using a Laplace transform and a Hankel transform, subsequently, and there I encountered a problem.
I'll jump right in, my equation is:
eqn = D[T[r, z, t], {r, 2}] + 1/r D[T[r, z, t], r] +
D[T[r, z, t], {z, 2}] == 1/[Alpha] D[T[r, z, t], t]
lapl = LaplaceTransform[eqn,t,p] gives me a ,what I understand is a, correct result:
LaplaceTransform[(T^(0,2,0))[r,z,t],t,p]+LaplaceTransform[(T^(1,0,0))[r,z,t],t,p]/r+
LaplaceTransform[(T^(2,0,0))[r,z,t],t,p]==
(p LaplaceTransform[T[r,z,t],t,p]-T[r,z,0])/[Alpha]
However, when applying HankelTransform[lapl,r,s,0] the result I'm getting is:
HankelTransform[LaplaceTransform[(T^(0,2,0))[r,z,t],t,p],r,s,0]+
HankelTransform[LaplaceTransform[(T^(1,0,0))[r,z,t],t,p]/r,r,s,0]+
HankelTransform[LaplaceTransform[(T^(2,0,0))[r,z,t],t,p],r,s,0]==
(1/[Alpha])(p HankelTransform[LaplaceTransform[T[r,z,t],t,p],r,s,0]-
HankelTransform[T[r,z,0],r,s,0])`
while, based on the paper, I'm supposed to get:
$$\frac{\mathrm d^2 \overline{T_0}}{\mathrm dz^2}-\left(s^2+\frac{p}{\alpha}\right)\overline{T_0}=0$$
(In case the output is to tiring to read, as summary: there is no occurrence of s whatsoever, which makes doubt the correctness.)
Do I use the functions in wrong way (which I suppose is more likely) or is Mathematica (11.2) not able to transform this kind of input. Thanks in advance and forgive me my poor formatting skills.
