4.3.1 Spherical coordinates [275] No angle dependencies [275] No angle dependencies

problem number 275

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 4.174:PDE specification


\[\left \{\left \{u(r,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {2 (-1)^{K[1]} \left (1-e^{-k \pi ^2 t K[1]^2}\right ) \sin (\pi r K[1])}{k \pi ^3 r K[1]^3}+t\right \}\right \}\]


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