@farahnaz The names _C1, _C2, etc. are used by pdsolve (and dsolve) as names of arbitrary constants.

In your case this would imply that you should determine the constants so that sol3[3] satisfies the requirements for the behavior in rho at 0 and infinity. That may be very difficult or practically impossible.

By using the laplace transform or by separating variables it appears that the solution can be written
T(z,t) = sin(Pi*z)*f(t)

where f(t) satisfies the following integral equation:
eq:=simplify(eval(PDE,{T=((z,t)->sin(Pi*z)*f(t))})/sin(Pi*z));
    
         

The initial values must be:
ics:={f(0)=1,D(f)(0)=0}; #Based on the initial values for T(z,t)

I tried with my own unsophisticated solver to find an approximate solution to eq with ics first in terms of monomials in t and on the interval 0..10. Since that produced ever increasing waves when animating T(z,t) I turned to using cos(n*Pi*t), n=0..N instead of the monomials.
I shall return if I get anything I believe could be close to an acceptable approximate solution.

@farahnaz When changing from dsolve to pdsolve I get what I (to great length) described.
As an example sol3[1] is found to be

H(rho, z) = (1/2)*Pi*(_C1*BesselJ(1, rho)+_C2*StruveH(1, rho))/_C2

which (as I also wrote) is independent of z.

I just edited my answer to include a correction: sol3[1] doesn't satisfy the requirement: I had them turned around.

Since sol3[2] doesn't either, you are stuck with sol3[3] which is the complicated one.

@tomleslie I just tried your code in Maple 12 after having replaced all occurrences of numelems with nops.
It worked nicely!
So surely it will work in Maple 13 too.

@shakuntala You don't update from Maple 13 to a newer version: You simply buy another version, so buy Maple 2016 since money is not a problem. Go to the Maplesoft website for purchase.

@torabi The command is what you used and what I used in the form
invlaplace(rhs(sol3),s,t);

It was not successful, however. invlaplace couldn't do the inversion, but just used linearity to get

Why is the title "Shooting Method" ?
There is no shooting going on. `dsolve/numeric/bvp`doesn't use a shooting method.

@Bendesarts Using your condition1 I tried

indets(condition1,`^` );

There are no squared sine functions among the members of the output:

@Christopher2222 With few exceptions (e.g. Edgardo S. Cheb-Terrab and Robert J. Lopez), Maple employees rarely participate on MaplePrimes, but leave the discussion to the members.
I'm sure there are very good reasons for that, among those

(1) too busy working on matters directly related to Maple software;
(2) allowing a free discussion among members without interfering with an "authoritative" or "official" verdict on the issue.

Both of those two are perfectly legitimate and honorable reasons.

But the spam issue is different as it has nothing to do with the behavior of Maple software, so some reaction from the responsible part would be appreciated.

@9009134 Why not keep the RootOf. Notice that it is indexed with 1,2,and 3.
Try e.g. setting all parameters to 1:

nms:=indets(A,name);
eval([Sol],nms=~1);
allvalues(%);
evalf(%); #If you want floats you can skip the previous step.

I didn't even think of trying to integrate u1 symbolically before, but doing that improves the result.
For convenience I bring all of the code.
I have made two extra changes. In series I have used the order 5 instead of 3.
Also it seems to help to use the procedural form for input to evalf/Int:

restart;
Digits:=15:
tD := 20;
u:=4/Pi^2*( 1 - exp(-x^2*tD) ) / ( x^3*( BesselJ(0,x)^2 + BesselY(0,x)^2 ) );
series(u,x,5); #Replaced 3 by 5
u1:=convert(%,polynom);
U1:=int(u1,x=0..1/2);
#evalf(Int(u-u1,x=0..1/2,epsilon=1e-6));
evalf(Int(unapply(u-u1,x), 0..1/2, epsilon=1e-12));
evalf(U1);
res1:=%%+%;
res2:=evalf(Int(u,x=1/2..infinity));
res1+res2;
simplify(%); # Result12.3201290912317

@Carl Love Yes, I would bulk delete.

The problem now is that the time it takes to delete N items is proportional to N.

Way most of the spam can be seen as such just from the title, but a bulk delete function could show the contents of the selected items so a quick eye scan can eliminate any doubt before deletion.
Real obvious items having titles like the one I'm staring at to the right of me http://testoteronebooster...etc...etc could as you suggest be deleted automatically.

Today Sept. 20 at 1 p.m. (Central European Summer Time) I counted 31 spam posts after clicking Recent Posts.

No way I'm going to start deleting those. I give up.

@Christopher2222 I sometimes wonder why somebody is filling this site with this cr.p, which nobody here can be expected to be interested in.
Is it simply because it is so easy to do and costs nothing (or close to nothing) and that maybe one in a thousand will bite?

I think this site relies on the users to remove the spam. That is OK as long as there is little. But that is not (or no more) the case.
The introduction of the feature "Delete as Spam" hasn't been helping anything it seems.

I delete quite a lot, but get tired of it besides irritated. You have to delete one at a time, which may seem reasonable, but the process is a two step procedure, so deleting 10-20 posts ("questions") is boring at best.
Could some of us delete several at a time then it might be tolerable.

Actually I suggested deleting anything that contains http in the title some time ago.

It would be nice if one didn't have to wake up to 10-20 spam "questons" or posts about weight loss or penis extensions or whatever. Why doesn't MaplePrimes have a spam filter?
It is irritating to have to delete that much spam or more before you can see anything of value.