3.3.1 Spherical coordinates

3.3.1.1 [273] No angle dependencies

problem number 273

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 \(u(1,t)=t\)

Figure 3.173: PDE specification

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'));
 

\[u \left (r , t\right ) = \frac {\mathcal {L}^{-1}\left (\frac {\operatorname {csch}\left (\frac {\sqrt {s}}{\sqrt {k}}\right ) \sinh \left (\frac {\sqrt {s}\, r}{\sqrt {k}}\right )}{s^{2}}, s , t\right )}{r}\]

Has unresolved Laplace integrals

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