This is a Wave PDE inside disk. with fixed edge of disk,  and no theta dependency. initial position and velocity  given.

When using "c^2" for the  wave propagation speed, Maple only gives solution when also using assumptions to tell that c is positive.

restart; 
Physics:-Version()[2]; 
`2018, November 28, 1:35 hours, version in the MapleCloud: 224, version installed in this computer: 224`

And the PDE is 

pde := diff(u(r, t), t$2) = c^2*( diff(u(r,t), r$2)  + (1/r)* diff(u(r,t),r)  ) ; 
ic:=u(r,0)=1, eval( diff(u(r,t),t),t=0)=r/3; 
bc:=u(1,t)=0; 
sol:=pdsolve([pde, ic,bc], u(r, t)) assuming t>0;

sol:=()

Now adding assuming c>0, or "c::positive", or "c>=0", it solves it 

restart;
pde := diff(u(r, t), t$2) = c^2*( diff(u(r,t), r$2)  + (1/r)* diff(u(r,t),r)  ) ; 
ic:=u(r,0)=1, eval( diff(u(r,t),t),t=0)=r/3; 
bc:=u(1,t)=0; 
sol:=pdsolve([pde, ic,bc], u(r, t)) assuming t>0,c>=0;

Any idea why the assumption "c>0"  is needed when the speed is given as "c^2" ? I also tried assumption "c::real", but it still did not solve it.   It seems related to Maple using Laplace transform to solve it.

It seems to me the assumption c>0 should not needed here. But if it is, I'd like to learn why.

Maple 2018.2, windows 10

Update

Thanks to comment below

   " so u(r, 0)=1 (for all r), and u(1, t)=0 (for all t). So what is u(1,0)??? "

I should have added "r>0,r<1".  The reason I did not, is that I copied the code to solve this from Mathematica. In Mathematica, it did not need this assumption to solve it. It seems to depend on the method of solution used by Mathematica vs. Maple to cause this difference.

But now I see it helps to have it there. With the above assumption, now, there is no need to do any assumption on "c" at all and it gives solution. adding c>0 or t>0 or c>=0, makes no change now.

restart;
pde := diff(u(r, t), t$2) = c^2*( diff(u(r,t), r$2)  + (1/r)* diff(u(r,t),r)  ) ; 
ic:=u(r,0)=1, eval( diff(u(r,t),t),t=0)=r/3; 
bc:=u(1,t)=0; 
sol:=pdsolve([pde, ic,bc], u(r, t)) assuming t>0,r>0,r<1;

convert(...,Int) in Maple 2018.2 works for fourier, invfourier, laplace, but does not work for invlaplace.  

Why is that? Is there a workaround?

expr:=fourier(f(x), x, w):
convert(expr,Int);

expr:=invfourier(f(w), w, x):
convert(expr,Int);

expr:=laplace(f(s),s,t):
convert(expr,Int)

expr:=invlaplace(f(s),s,t):
convert(expr,Int)

Was expecting to see the Mellin's inverse formula.

Maple 2018.2 on windows 10

Using Latest Physics updates (I am not sure when this started), pdsolve gives Error, (in PDEtools/eval/2) numeric exception: division by zero on the following problem from a HW from text book.

restart;
PackageTools:-IsPackageInstalled("Physics Updates");
                             "220"


pde:=diff(w(x,t),t)+3*t*diff(w(x,t),x)=w(x,t);
ic:=w(x,0)=f(x);
sol:=pdsolve([pde,ic],w(x,t));

Error, (in PDEtools/eval/2) numeric exception: division by zero
 

Mathematica answer btw is 

pde = D[w[x, t], t] + 3 t D[w[x, t], x] == w[x, t];
ic = w[x, 0] == f[x];
sol = Simplify[DSolve[{pde, ic}, w[x, t], {x, t}]]

This is on Maple 2018.2 on windows 10 64 bit.

Any idea what is causing this and any workaround? Do others get the same exception?

Trying Maple on a textbook problem to verify my hand solution. 

But Maple pdsolve hangs with the mserver.exe tallomg almost 100% CPU and over 10 GB of RAM!

I waited for almost 20 minutes. Tried another time, same thing.

It is no problem if Maple can't solve this, but Maple seems to suffer from too many hangs when it is not able to solve a problem. I've had similar problems with dsolve also.

This is on windows 10, 64 bit with Maple 2018.2 With Physics updates version 218

restart;
PackageTools:-IsPackageInstalled("Physics Updates");

218

interface(showassumed=0);
infolevel[pdsolve]:=2;
pde :=  diff(u(x,t),t)=k*diff(u(x,t),x$2)+Q(x,t);
ic  :=  u(x,0)=f(x);
bc  :=  eval(diff(u(x,t),x),x=0)=A(t),eval(diff(u(x,t),x),x=1)=B(t);
sol:=pdsolve({pde,ic,bc},u(x,t)) assuming t>0,k>0;

then

* trying method "_Fn" for 2nd order PDEs
   -> trying "linear_in_xt"
   -> trying "BC_equal_0"
* trying method "_Cn_cn" for 2nd order PDEs
Trying travelling wave solutions as power series in tanh ...
Trying travelling wave solutions as power series in ln ...
Trying travelling wave solutions as power series in tanh ...
Trying travelling wave solutions as power series in ln ...
* trying method "Wave" for 2nd order PDEs
   -> trying "Cauchy"
   -> trying "SemiInfiniteDomain"
   -> trying "WithSourceTerm"
* trying method "Heat" for 2nd order PDEs
   -> trying "SemiInfiniteDomain"
   -> trying "WithSourceTerm"
* trying method "Series" for 2nd order PDEs
   -> trying "ThreeBCsincos"
   -> trying "FourBC"
   -> trying "ThreeBC"
   -> trying "ThreeBCPeriodic"
   -> trying "WithSourceTerm"
      * trying method "_Fn" for 2nd order PDEs
         -> trying "linear_in_xt"
         -> trying "BC_equal_0"
      * trying method "_Cn_cn" for 2nd order PDEs

And here is hangs. Notice that because both ends are Neumann, there is no unique solution to this problem.  So the solution will contain arbitrary constant. May be this is what made pdsolve hang? 

No it is not. Trying with only one end nonhomogeneous  Neumann, and the other end Dirichlet, it still hangs. The problem seems to be with one end is nonhomogeneous  Neumann, which is a function of time. So this hangs also (same place)

restart;
interface(showassumed=0);
infolevel[pdsolve]:=2;
pde :=  diff(u(x,t),t)=k*diff(u(x,t),x$2)+Q(x,t);
ic  :=  u(x,0)=f(x);
bc  :=  eval(diff(u(x,t),x),x=0)=A(t),eval(diff(u(x,t),x),x=1)=0;
sol:=pdsolve({pde,ic,bc},u(x,t)) assuming t>0,k>0;

And this also

restart;
interface(showassumed=0);
infolevel[pdsolve]:=2;
pde :=  diff(u(x,t),t)=k*diff(u(x,t),x$2)+Q(x,t);
ic  :=  u(x,0)=f(x);
bc  :=  eval(diff(u(x,t),x),x=0)=sin(t),eval(diff(u(x,t),x),x=1)=0;
sol:=pdsolve({pde,ic,bc},u(x,t)) assuming t>0,k>0;

But this does not hang

interface(showassumed=0);
infolevel[pdsolve]:=2;
pde :=  diff(u(x,t),t)=k*diff(u(x,t),x$2)+Q(x,t);
ic  :=  u(x,0)=f(x);
bc  :=  eval(diff(u(x,t),x),x=0)=1,eval(u(x,t),x=1)=0;
sol:=pdsolve({pde,ic,bc},u(x,t)) assuming t>0,k>0;

The issue seems to be when one end is nonhomogeneous  Neumann which is function of time. 

Is there a workaround so it does not hang? The complaint is that Maple hangs, and not that Maple unable to solve the PDE.

Why  

eval(diff(u(x,t),x),x=0)=A(t)

gives

But

eval(diff(u(x,t),x),x=L)=A(t)

gives

I was expecting the same syntax in both cases. It seems for numbers Maple uses the first syntax and for symbols it uses the second syntax.

Does one need to worry about this difference?