Added March 28, 2019.
Problem 1, section 41, Fourier series and boundary value problems 8th edition by Brown and Churchill.
Solve \(u_t = \nabla u \) where \(\nabla u = \frac {1}{r} (r u)_{rr} \) in Spherical coordinates with initial conditions \(u(r,0)=0\) and boundary conditions \(u(1,t)=t\)
Mathematica ✓
ClearAll["Global`*"]; pde = D[u[r, t], t] == (k*D[r*u[r, t], {r, 2}])/r; ic = u[r, 0] == 0; bc = u[1, t] == t; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde, ic, bc}, u[r, t], {r, t}, Assumptions -> {t > 0, k > 0}], 60*10]];
\[\left \{\left \{u(r,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {2 (-1)^{K[1]} \left (1-e^{-k \pi ^2 t K[1]^2}\right ) \sin (\pi r K[1])}{k \pi ^3 r K[1]^3}+t\right \}\right \}\]
Maple ✓
restart; pde := diff(u(r,t),t)= k/r*diff(r*u(r,t),r$2); ic := u(r,0)=0; bc := u(1,t) =t; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic,bc],u(r,t),HINT =boundedseries(r=0)) assuming t>0,k>0) ,output='realtime'));
\[u \left ( r,t \right ) ={\frac {1}{r}{\it invlaplace} \left ( {\frac {1}{{s}^{2}}\sinh \left ( {r\sqrt {s}{\frac {1}{\sqrt {k}}}} \right ) \left ( \sinh \left ( {\sqrt {s}{\frac {1}{\sqrt {k}}}} \right ) \right ) ^{-1}},s,t \right ) }\] Has unresolved Laplace integrals
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