#### 3.3.1 Spherical coordinates

3.3.1.1 [273] No angle dependencies

##### 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 $$u(1,t)=t$$

Mathematica

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

Maple

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

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